40s applied perm comb assign 2012
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
1. Eve can choose from the following notebooks:
• lined pages come in red, green, blue, and purple
• graph paper comes in orange and black
How many different colour variations can Eve choose if she needs one lined notebook and one
with graph paper?
A.
B.
C.
D.
____
6
8
12
16
2. Eve can choose from the following notebooks:
• lined pages come in red, green, blue, and purple
• graph paper comes in orange and black
If Eve needs one lined notebook and one with graph paper, which of the following pairs is not a
possible outcome?
A.
B.
C.
D.
____
3. A combination lock opens with the correct three-digit code. Each wheel rotates through the digits 1
to 8. How many different three-digit codes are possible?
A.
B.
C.
D.
____
red and orange
black and blue
green and red
purple and black
24
64
512
1024
4. A combination lock opens with the correct three-digit code. Each wheel rotates through the digits 1
to 8. Suppose each digit can be used only once in a code. How many different codes are possible
when repetition is not allowed?
A.
B.
C.
D.
21
63
256
336
____
5. A combination lock opens with the correct four-letter code. Each wheel rotates through the letters
A to L. How many different four-letter codes are possible?
A.
B.
C.
D.
____
6. A combination lock opens with the correct four-letter code. Each wheel rotates through the letters
A to L. Suppose each letter can be used only once in a code. How many different codes are
possible when repetition is not allowed?
A.
B.
C.
D.
____
3481
3540
3600
3660
9. A restaurant offers 60 flavours of wings and your choice of three dips. How many variations of
wings and dip can you order?
A.
B.
C.
D.
____
3481
3540
3600
3660
8. A restaurant offers 60 flavours of wings. How many ways can two people order two servings of
wings, either the same flavour or different flavours?
A.
B.
C.
D.
____
20 736
11 880
1320
8976
7. A restaurant offers 60 flavours of wings. How many ways can two people order two different
flavours?
A.
B.
C.
D.
____
20 736
48
1728
456 976
20
60
180
216 000
10. The lunch special at a sandwich bar offers you a choice of 6 sandwiches, 4 salads, 6 drinks, and 3
desserts. How many different meals are possible if you choose one item from each category?
A.
B.
C.
D.
432
576
646
720
____
11. The dinner special at a restaurant offers you a choice of 8 entrees, 2 salads, 5 drinks, and 3
desserts. How many different meals are possible if you choose one item from each category?
A.
B.
C.
D.
____
12. The lunch special at a diner offers you a choice of 5 sandwiches, 2 salads, 3 soups, 6 drinks, and 2
desserts. How many different meals are possible if you choose one item from each category?
A.
B.
C.
D.
____
13
20
21
26
15. The numbers 1 to 20 are written on slips of paper and put in a hat. How many possible ways can
you draw a either an odd number or a two-digit number from the hat?
A.
B.
C.
D.
____
2
13
14
26
14. How many possible ways can you draw a single card from a standard deck and get an even
number?
A.
B.
C.
D.
____
432
360
526
720
13. How many possible ways can you draw a single card from a standard deck and get either a heart or
a club?
A.
B.
C.
D.
____
360
380
420
480
13
14
15
16
16. Evaluate.
8! + 1!
A.
B.
C.
D.
40 321
5041
40 123
16 777 217
____
17. Evaluate.
(3!)2
A.
B.
C.
D.
____
8
9
18
36
18. Evaluate.
A. 0
B. 1
C. 3
D.
____
19. Evaluate.
A.
B.
C.
D.
____
20. Evaluate.
A.
B.
C.
D.
____
13
16
20
23
21. Identify the expression that is equivalent to the following:
A.
B.
C.
D.
____
1 000 000
1 001 000
10 100 100
999 999
n
–n
n2
n3
22. Identify the expression that is equivalent to the following:
A.
B.
C. n2
D. n!
____
23. Identify the expression that is equivalent to the following:
A.
B.
C. n3
D. (n + 1)!
____
24. Solve for n, where n  I.
A.
B.
C.
D.
____
25. Solve for n, where n  I.
A.
B.
C.
D.
____
8
16
24
32
26. Solve for n, where n  I.
A.
B.
C.
D.
____
10
19
20
39
13
15
17
18
27. Solve for n, where n  I.
A.
B.
C.
D.
____
28. Solve for n, where n  I.
A.
B.
C.
D.
____
48
120
720
24
31. Evaluate.
3P1
A.
B.
C.
D.
____
49
128
720
5040
30. How many different permutations can be created when Anneliese, Becky, Carlo, Dan, and Esi line
up to buy movie tickets, if Esi always stands immediately behind Becky?
A.
B.
C.
D.
____
8
9
10
11
29. How many different permutations can be created when 7 people line up to buy movie tickets?
A.
B.
C.
D.
____
3
4
5
6
1
2
3
6
32. Evaluate.
21P2
A.
B.
C.
D.
441
420
399
2 097 152
____
33. Evaluate.
14P7
A.
B.
C.
D.
____
34. Evaluate.
200P0
A.
B.
C.
D.
____
72
100
81
90
38. How many numbers are there from 1000 to 1999 that do not have any repeated digits?
A.
B.
C.
D.
____
78 125
16 807
2520
1250
37. Suppose a word is any string of letters. How many two-letter words can you make from the letters
in LETHBRIDGE if you do not repeat any letters in the word?
A.
B.
C.
D.
____
20
16
216
120
36. Suppose a word is any string of letters. How many five-letter words can you make from the letters
in KELOWNA if you do not repeat any letters in the word?
A.
B.
C.
D.
____
0
1
2
200
35. Suppose a word is any string of letters. How many three-letter words can you make from the letters
in REGINA if you do not repeat any letters in the word?
A.
B.
C.
D.
____
17 297 280
2 162 160
121 080 960
105 413 504
504
1000
888
776
39. How many numbers are there from 900 to 999 that do not have any repeated digits?
A.
B.
C.
D.
____
40. Solve for n.
nP4 = 120
A.
B.
C.
D.
____
r=5
r=6
r=1
r=3
44. How many ways can 7 friends stand in a row for a photograph if Sheng always stands beside his
girlfriend?
A.
B.
C.
D.
____
r=1
r=2
r=3
r=4
43. Solve for r.
15Pr – 2 = 2730
A.
B.
C.
D.
____
n=5
n=6
n=7
n=8
42. Solve for r.
9Pr = 72
A.
B.
C.
D.
____
n=5
n=6
n=7
n=8
41. Solve for n.
n – 2P2 = 30
A.
B.
C.
D.
____
81
90
100
72
1440
5040
360
720
45. How many ways can 8 friends stand in a row for a photograph if Molly, Krysta, and Simone
always stand together?
A.
B.
C.
D.
____
46. Evaluate.
A.
B.
C.
D.
____
30 030
30 300
60 060
60 600
49. Evaluate.
A.
B.
C.
D.
____
840
6720
13 440
1680
48. Evaluate.
A.
B.
C.
D.
____
48
72
140
180
47. Evaluate.
A.
B.
C.
D.
____
1440
4320
5040
2160
330
660
990
1320
50. How many different arrangements can be made using all the letters in CANADA?
A.
B.
C.
D.
120
180
360
720
____
51. How many different arrangements can be made using all the letters in NUNAVUT?
A.
B.
C.
D.
____
52. How many different arrangements can be made using all the letters in ATHABASCA?
A.
B.
C.
D.
____
360
480
120
720
54. How many different routes are there from A to B, if you only travel south or east?
A.
B.
C.
D.
____
60 480
10 080
15 120
90 720
53. How many different arrangements can be made using all the letters in CALGARY, if the first letter
must be G?
A.
B.
C.
D.
____
630
1260
2520
5040
16
24
28
56
55. How many different routes are there from A to B, if you only travel south or east?
A. 10
B. 20
C. 40
D. 8
____
56. How many different routes are there from A to B, if you only travel north or east?
A.
B.
C.
D.
____
57. How many different routes are there from A to B, if you only travel south or east?
A.
B.
C.
D.
____
128
256
156
104
58. Five quarters are flipped simultaneously. How many ways can three coins land heads and two
coins land tails?
A.
B.
C.
D.
____
100
250
400
350
12
10
15
5
59. Eight quarters are flipped simultaneously. How many ways can three coins land heads and five
coins land tails?
A. 36
B. 42
C. 50
D. 56
____
60. Eight quarters are flipped simultaneously. How many ways can at least six coins land heads?
A.
B.
C.
D.
____
61. There are 14 members of a student council. How many ways can 4 of the members be chosen to
serve on the dance committee?
A.
B.
C.
D.
____
110
220
330
440
64. The numbers 10 to 16 are written on identical slips of paper and put in a hat. How many ways can
2 numbers be drawn simultaneously?
A.
B.
C.
D.
____
1144
1716
3432
17 297 280
63. The numbers 1 to 11 are written on identical slips of paper and put in a hat. How many ways can
4 numbers be drawn simultaneously?
A.
B.
C.
D.
____
1001
2002
6006
24 024
62. There are 14 members of a student council. How many ways can 7 of the members be chosen to
serve on the dance committee?
A.
B.
C.
D.
____
36
37
44
56
21
15
30
42
65. A fun fair requires 4 employees to work at the sack bar. There are 13 people available. How
many ways can a group of 4 be chosen?
A.
B.
C.
D.
1000
715
635
808
____
66. Evaluate.
A.
B.
C.
D.
____
0
1
11
22
15
18
30
36
70. How many ways can 3 representatives be chosen from a soccer team of 16 players?
A.
B.
C.
D.
____
118
69. Evaluate.
A.
B.
C.
D.
____
130
126
122
68. Evaluate.
A.
B.
C.
D.
____
16
67. Evaluate.
A.
B.
C.
D.
____
0
1
4
1120
560
3360
1580
71. How many ways can 4 representatives be chosen from a hockey team of 17 players?
A.
B.
C.
D.
2380
57 120
31 060
9575
____
72. How many ways can 2 representatives be chosen from a class of 28 students?
A.
B.
C.
D.
____
73. Suppose that 3 teachers and 6 students volunteered to be on a graduation committee. The
committee must consist of 1 teachers and 2 students. How many different graduation committees
does the principal have to choose from?
A.
B.
C.
D.
____
45
60
90
180
74. Suppose that 10 teachers and 8 students volunteered to be on an environmental action committee.
The committee must consist of 2 teachers and 2 students. How many different environmental
action committees does the principal have to choose from?
A.
B.
C.
D.
____
1512
7560
756
378
45
73
1260
5040
75. Which of the following is equivalent to
?
A.
B.
C.
D.
____
76. Which of the following is equivalent to
A.
B.
C.
?
D.
____
77. Which of the following is equivalent to
?
A.
B.
C.
D.
____
78. Which of the following is equivalent to
?
A.
B.
C.
D.
____
79. Solve for n.
nC1 = 30
A.
B.
C.
D.
____
80. Solve for r.
10Cr = 45
A.
B.
C.
D.
____
n=6
n = 10
n = 30
n = 60
r=2
r=5
r=8
A and C
81. Identify the term that best describes the following situation:
Determine the number of arrangements of six friends waiting in line for movie tickets.
A.
B.
C.
D.
permutations
combinations
factorial
none of the above
____
82. Identify the term that best describes the following situation:
Determine the number of codes for a lock with three dials numbered 0 to 9.
A.
B.
C.
D.
____
83. Identify the term that best describes the following situation:
Determine the number of pizzas with 4 different toppings from a list of 40 toppings.
A.
B.
C.
D.
____
permutations
combinations
factorial
none of the above
86. How many ways can the 6 starting positions on a hockey team (1 goalie, 2 defense, 3 forwards) be
filled from a team of 1 goalie, 4 defense, and 8 forwards?
A.
B.
C.
D.
____
permutations
combinations
factorial
none of the above
85. Identify the term that best describes the following situation:
Determine the number of two-card hands you can be dealt from a standard deck of 52 cards.
A.
B.
C.
D.
____
permutations
combinations
factorial
none of the above
84. Identify the term that best describes the following situation:
Determine the number of ways three horses can finish first, second, and third in a race of
12 horses.
A.
B.
C.
D.
____
permutations
combinations
factorial
none of the above
164
254
336
1716
87. How many ways can the 6 starting positions on a hockey team (1 goalie, 2 defense, 3 forwards) be
filled from a team of 2 goalies, 4 defense, and 7 forwards?
A.
B.
C.
D.
420
500
858
1716
____
88. How many ways can the 6 starting positions on a hockey team (1 goalie, 2 defense, 3 forwards) be
filled from a team of 2 goalies, 5 defense, and 10 forwards?
A.
B.
C.
D.
____
89. Ten friends on bicycles find an empty bike rack with exactly ten spots for bikes. How many ways
can the ten bikes be parked so that Elsa and Rashid are parked next to each other?
A.
B.
C.
D.
____
1200
2400
4800
9600
362 880
725 760
2 177 280
2 000 000 000
90. Ten friends on bicycles find an empty bike rack with exactly ten spots for bikes. How many ways
can the ten bikes be parked so that Elsa is at one end of the bike rack?
A. 9!
B.
C. 10!
D.
____
91. Ten friends on bicycles find an empty bike rack with exactly ten spots for bikes. How many ways
can the ten bikes be parked so that Elsa and Rashid are at either end of the bike rack?
A.
B.
C.
D.
____
92. Euchre is played with a deck of 24 cards that is similar to a standard deck of 52 playing cards, but
with only the ace, 9, 10, jack, queen, and king for all four suits.
How many different four-card hands are there with one card from each suit?
A.
B.
C.
D.
____
80 640
161 280
322 560
1 814 400
1248
10 626
12 480
1296
93. Euchre is played with a deck of 24 cards that is similar to a standard deck of 52 playing cards, but
with only the ace, 9, 10, jack, queen, and king for all four suits.
How many different four-card hands are there with at least three hearts?
A. 375
B. 926
C. 3336
D. 10 626
____
94. Euchre is played with a deck of 24 cards that is similar to a standard deck of 52 playing cards, but
with only the ace, 9, 10, jack, queen, and king for all four suits.
How many different five-card hands are there with at least three clubs?
A.
B.
C.
D.
____
375
926
3336
10 626
95. From a standard deck of 52 cards, how many different five-card hands are there with at least four
black cards?
A.
B.
C.
D.
388 700
649 740
1 299 480
454 480
Short Answer
1. Indicate whether the Fundamental Counting Principle applies to this situation:
Counting the number of possibilities when drawing a face card from a standard deck.
2. Indicate whether the Fundamental Counting Principle applies to this situation:
Counting the number of possibilities to choose from when buying a bicycle available in 4 sizes and
3 colours.
3. Indicate whether the Fundamental Counting Principle applies to this situation:
Counting the number of possibilities when picking a chair and a vice chair from a list of committee
members.
4. A band sells shirts and CDs at their concerts. They have 3 CDs and there are 4 different styles of
shirt available in small, medium, large, and extra large.
How many ways could you buy one CD and one shirt if you only consider one size of shirt?
5. A band sells shirts and CDs at their concerts. They have 3 CDs and there are 4 different styles of
shirt available in small, medium, large, and extra large.
How many ways could someone buy two different CDs and a shirt?
6. A band sells shirts and CDs at their concerts. They have 5 CDs and there are 8 different styles of
shirt available in 5 sizes.
How many ways could someone buy a CD and a shirt?
7. A combination lock opens with the correct three-letter code. Each wheel rotates through the letters
A to O. How many different three-letter codes are possible?
8. A combination lock opens with the correct three-letter code. Each wheel rotates through the letters
A to O. Suppose each letter can be used only once in a code. How many different codes are
possible when repetition is not allowed?
9. The “Pita Patrol” offers these choices for each sandwich:
• white or whole wheat pitas
• 3 types of cheese
• 5 types of filling
• 12 different toppings
• 4 types of sauce
How many different pitas can be made with 1 cheese, 1 filling, 1 topping, and no sauce?
10. The “Pita Patrol” offers these choices for each sandwich:
• white or whole wheat pitas
• 3 types of cheese
• 5 types of filling
• 12 different toppings
• 4 types of sauce
How many different pitas can be made with 1 cheese, 1 filling, 1 topping, and 1 sauce?
11. The “Pita Patrol” offers these choices for each sandwich:
• white or whole wheat pitas
• 3 types of cheese
• 5 types of filling
• 12 different toppings
• 4 types of sauce
How many different pitas can be made with no cheese, 1 filling, 2 different toppings, and 1 sauce?
12. A theatre is showing 3 action movies, 4 comedies, 4 dramas, and 1 foreign film. How many
choices does Sophia have if she does not want to watch a drama or the foreign film?
13. A theatre is showing 2 action movies, 3 comedies, 3 dramas, 2 horror movies, and 2 foreign films.
How many choices does Sophia have if she does not want to watch an action movie or a horror
movie?
14. The numbers 1 to 20 are written on slips of paper and put in a hat. How many possible ways can
you draw a either a number less than 5 or a perfect square from the hat?
15. The numbers 1 to 20 are written on slips of paper and put in a hat. How many possible ways can
you draw a either a prime number or a multiple of 6 from the hat?
16. Evaluate.
17. Evaluate.
11  10  9!
18. Evaluate.
19. Evaluate.
4! 3!  2!
20. Evaluate.
21. Write the following expression using factorial notation.
765432
22. Write the following expression using factorial notation.
87654
23. Write the following expression using factorial notation.
24. Write the following expression using factorial notation.
25. How many ways can you arrange the letters in the word FACTOR?
26. A baseball coach is determining the batting order for the nine players she is fielding. The coach
has already decided who will bat first and second. How many different batting orders are possible?
27. Solve for n, where n  I.
28. Solve for n, where n  I.
29. Solve for n, where n  I.
30. Solve for n, where n  I.
31. Evaluate.
6P5
32. Evaluate.
5P3
33. Evaluate.
100P1
34. Evaluate.
12P5
35. Without calculating, predict which value is larger:
100P70 or 100P50
36. Without calculating, predict which value is larger:
12P10 or 10P8
37. There are nine different marbles in a bag. Suppose you reach in and draw one at a time, and do this
three times. How many ways can you draw the three marbles if you do not replace the marble each
time?
38. There are twelve different marbles in a bag. Suppose you reach in and draw two marbles one at a
time without replacement. How many ways can you draw the two marbles?
39. There are nine different marbles in a bag. Suppose you reach in and draw one at a time, and do this
six times. How many ways can you draw the six marbles if you do not replace the marble each
time?
40. Solve for n.
nP2 = 90
41. Solve for n.
nP3 = 1320
42. Solve for r.
34Pr = 34
43. Solve for r.
8Pr
= 1680
44. Solve for r.
5(6Pr) = 600
45. If n is a positive integer, then what are the values of nP0, nP1, and nPn?
46. Evaluate.
47. Evaluate.
48. Evaluate.
49. Evaluate.
50. How many different arrangements can be made using all the letters in WINNIPEG?
51. How many different arrangements can be made using all the letters in VANCOUVER?
52. How many different arrangements can be made using all the letters in YELLOWKNIFE?
53. How many different arrangements can be made using all the letters in YELLOWKNIFE, if the first
letter must be L and the last letter must be Y?
54. How many different routes are there from A to B, if you only travel south or east?
55. How many different routes are there from A to B, if you only travel south or east?
56. How many different routes are there from A to B, if you only travel south or east?
57. How many different routes are there from A to B, if you only travel north or east?
58. A true-false test has twelve questions. How many different permutations of answers can the
teacher create if half of the answers are true and half of the answers are false?
59. A true-false test has ten questions. How many different permutations of answers can the teacher
create if six answers are true and four answers are false?
60. A true-false test has ten questions. How many different permutations of answers can the teacher
create if at least seven answers are true?
61. There are 12 members of a student council. How many ways can 5 of the members be chosen to
serve on the dance committee?
62. The numbers 1 to 16 are written on identical slips of paper and put in a hat. How many ways can 2
numbers be drawn simultaneously?
63. The numbers 10 to 18 are written on identical slips of paper and put in a hat. How many ways can
7 numbers be drawn simultaneously?
64. A fun fair requires 6 employees to help move one of the booths. There are 8 people available. How
many ways could a group of 6 be chosen?
65. A fun fair requires 3 employees to sell tickets. There are 9 people available. How many ways could
a group of 3 be chosen?
66. Evaluate.
67. Evaluate.
68. Evaluate.
69. Evaluate.
70. How many ways can you select 2 different flavours of ice-cream for a sundae if there are
16 flavours available?
71. How many ways can you select 3 different flavours of ice-cream for a sundae if there are
31 flavours available?
72. The numbers 1 to 49 are written on identical slips of paper and put in a bag. How many different
ways can you reach in the bag and pull out 6 numbers at once?
73. Tad is selecting music for a long car trip. Suppose he has 10 rock albums and 12 hip-hop albums.
How many different ways can he select 1 rock album and 2 hip-hop albums?
74. A Bingo cage contains 4 different B balls and all 15 N balls. Suppose Veronica reaches in grabs 2
B balls and 3 N balls. How many different combinations of these balls are there?
75. How many 5-person committees can be formed from a group of 8 teachers and 5 students if there
must be exactly 3 students on the committee?
76. How many 4-person committees can be formed from a group of 8 teachers and 5 students if there
must be either 1 or 2 teachers on the committee?
77. Solve for n.
n + 4C1 = 14
78. Solve for n.
2nC2
= 120
79. Solve for r.
34Cr = 1
80. Solve for r.
6Cr = 20
81. Eight friends on bicycles find an empty bike rack with exactly eight spots for bikes. How many
ways can the eight bikes be parked so that Adya and Moira are parked next to each other?
82. Eight friends on bicycles find an empty bike rack with exactly eight spots for bikes. How many
ways can the eight bikes be parked so that Adya and Moira are parked at the ends of the rack?
83. A physics teacher has three topics for students to research: baryons, mesons, and leptons. How
many different ways can her class of 24 students be divided evenly among the 3 topics?
84. A physics teacher has four topics for students to research: reflection, refraction, the visible
spectrum, and the speed of light. How many different ways can her class of 24 students be divided
evenly among the 4 topics?
85. A phys-ed teacher needs 4 equal teams for an activity. How many different ways can her class of
24 students be divided evenly into 4 teams?
86. A phys-ed teacher needs 3 equal teams for an activity. How many different ways can his class of
21 students be divided evenly into 3 teams?
87. Euchre is played with a deck of 24 cards that is similar to a standard deck of 52 playing cards, but
with only the ace, 9, 10, jack, queen, and king for all four suits.
How many different four-card hands are there with no aces?
88. Euchre is played with a deck of 24 cards that is similar to a standard deck of 52 playing cards, but
with only the ace, 9, 10, jack, queen, and king for all four suits.
How many different four-card hands are there with at least two aces?
89. Euchre is played with a deck of 24 cards that is similar to a standard deck of 52 playing cards, but
with only the ace, 9, 10, jack, queen, and king for all four suits.
How many different five-card hands are there with at least four red cards?
90. From a standard deck of 52 cards, how many different three-card hands are there with at most one
ace?
91. From a standard deck of 52 cards, how many different four-card hands are there with at most two
diamonds?
92. How many different routes can you take to get to a restaurant eight blocks north and two blocks
east, if you travel only north or east?
93. How many different routes can you take to get to a restaurant six blocks north and six blocks west,
if you travel only north or west and start by walking north?
94. How many ways can the top four cash prices be awarded in a lottery that sold 150 tickets if each
ticket is replaced when drawn?
95. How many ways can the top four cash prices be awarded in a lottery that sold 150 tickets if each
ticket is not replaced when drawn?
Problem
1. A die is rolled and a coin is tossed.
a) Draw a tree diagram to determine how many outcomes are possible.
b) Confirm your answer to part a) using the Fundamental Counting Principle.
2. Hannah plays on a local hockey team. The hockey uniform has:
• four different sweaters: white, blue, grey, and black, and
• two different pants: blue and grey.
a) Draw a tree diagram to determine how many different variations of the uniform the coach can
choose from for each game are possible.
b) Confirm your answer to part a) using the Fundamental Counting Principle.
3. Xtreme clothing company makes ski jackets in three colours (yellow, red, and silver) and sizes of
extra small, small, medium, large, and extra large.
a) Draw a tree diagram to determine how many different colour–size variations of ski jackets the
company makes.
b) Confirm your answer to part a) using the Fundamental Counting Principle.
4. A combination lock opens with the correct four-digit code. Each wheel rotates through the digits 1
to 8.
a) How many different four-digit codes are possible?
b) Suppose each number can be used only once in a code. How many different codes are possible
when repetition is not allowed?
5. The locks on a briefcase open with the correct six-digit code. Each wheel rotates through the digits
0 to 9.
a) How many different six-digit codes are possible?
b) What percent of these codes have no repeated digits? Give your answer to the nearest percent.
6. A combination lock opens with the correct three-letter code. Each wheel rotates through the letters
A to T.
a) Suppose each letter can be used only once in a code. How many different codes are possible
when repetition is not allowed?
b) How many more codes would there be if repetition is allowed?
7. A combination lock opens with the correct four-letter code. Each wheel rotates through the letters
A to L.
a) How many different four-letter codes are possible?
b) What percent of these codes repeat at least one letter? Give your answer to the nearest percent.
8. Euchre is played with a deck of 24 cards that is similar to a standard deck of 52 playing cards, but
with only the ace, 9, 10, jack, queen, and king for all four suits.
Count the number of possibilities of drawing a single card from a euchre deck and getting:
a) either a jack or a red ace
b) either a number card (9 or 10) or a heart
9. Euchre is played with a deck of 24 cards that is similar to a standard deck of 52 playing cards, but
with only the ace, 9, 10, jack, queen, and king for all four suits.
a) Count the number of possibilities of drawing a single card from a euchre deck and getting either
a face card or a red card.
b) Does the Fundamental Counting Principle apply to this situation? Explain.
10. A standard deck contains 52 cards. A four-sided die has the numbers 1 to 4 on its sides. Suppose
you roll the die and draw a single card from the deck.
Count the number of possibilities of rolling the die and drawing either a face card or a spade. Show
your work.
11. Evaluate the following. Show your work.
12. Evaluate the following. Show your work.
13. Can you evaluate the following? Explain how you know.
(–2)!
14. Can you evaluate the following? Explain how you know.
15. Consider the word NUMBERS and all the ways you can arrange its letter using each letter only
once.
a) One possible permutation is ENBRUMS. Write three other possible permutations.
b) Use factorial notation to represent the total number permutations possible. Explain why your
expression makes sense.
16. Consider the word PERMUTE and all the ways you can arrange its letter using each letter only
once if the first letter is E.
a) One possible permutation is EPRETUM. Write three other possible permutations.
b) Use factorial notation to represent the total number permutations possible. Explain why your
expression makes sense.
17. Which value is greater? Show your work.
A. 6!
B.
18. Which value is greater? Show your work.
A.
B.
19. Which value is greater? Show your work.
A. 10!
B.
20. A vacation resort offers 3 different morning activities and 4 different afternoon activities for
visitors. Suppose you wanted to try all 7 activities in one day. Use the Fundamental Counting
Principle to determine how many different orders are possible, if all 3 morning activities must be
completed first. Show your work.
21. At a used car lot, 8 different car models are to be parked close to the street for easy viewing, but
there is only space for 6 cars. How many ways can 6 of the 8 cars be parked in a row? Show your
work.
22. At a used car lot, 11 different car models are to be parked close to the street for easy viewing, but
there is only space for 5 cars. How many ways can 5 of the 11 cars be parked in a row? Show your
work.
23. At a used car lot, 5 different car models are to be parked close to the street for easy viewing. The
lot has 4 red cars and 6 silver cars for the display. How many ways can the 5 cars be parked, if 2
red cars must be parked at either end of a row of 3 silver cars? Show your work.
24. At a used car lot, 6 different car models are to be parked close to the street for easy viewing. The
lot has 3 red cars and 8 silver cars for the display. How many ways can the 6 cars be parked, if 2
red cars must be parked at either end of a row of 4 silver cars? Show your work.
25. Mo has 12 new songs on his mp3 player. How many different 5-song playlists can be created from
his new songs, if no songs are repeated? Show your work.
26. Mo has 140 songs on his mp3 player.
a) How many different possibilities are there for the first three songs he hears, if he sets the player
to play the songs in random order without repeating? Show your work.
b) How does your answer change if repeating songs is allowed?
27. Salima has 172 songs on her mp3 player.
a) How many different possibilities are there for the first four songs she hears, if she sets the player
to play the songs in random order without repeating? Show your work.
b) How does your answer change if repeating songs is allowed?
28. An isogram is a word or phrase without a repeating letter.
Two of the longest one-word isograms in English are UNCOPYRIGHTABLE and
AMBIDEXTROUSLY.
a) How many ways can you make 3-letter initials from UNCOPYRIGHTABLE, if repeating letters
is not allowed? Show your work.
b) How many ways can you make 3-letter initials from AMBIDEXTROUSLY, if repeating letters
is allowed? Show your work.
c) How do the values for parts a) and b) compare?
29. An isogram is a word or phrase without a repeating letter.
Vito and Kira are playing a guessing game involving isograms. Vito thinks of a word with no
repeated letters. He tells Kira that his word can be used to make 42 letter pairs. He gives LS, OG,
and GO as examples.
a) How many letters are in Vito’s word?
b) What could Vito’s word be?
30. An isogram is a word or phrase without a repeating letter.
Vito and Kira are playing a guessing game involving isograms. Kira thinks of a word with no
repeated letters. She tells Vito that her word can be used to make 100 one- or two-letter phrases,
without repetition. She gives A, ET, and TE as examples.
a) How many letters are in Kira’s word? Show your work.
b) Which of the following could be Kira’s word? Explain your answer.
Switzerland atmospheric lumberjack duplicate
trapezoid
juxtaposes
31. A conveyor belt sushi restaurant lets you choose what to eat from different-coloured plates that go
by. You pay based on the colours of the plates.
After lunch, Ming has 2 red plates, 4 yellow plates, and 1 blue plate. How many ways can she
stack her plates in a single tower? Show your work.
32. A conveyor belt sushi restaurant lets you choose what to eat from different-coloured plates that go
by. You pay based on the colours of the plates.
After dinner, Claire has 3 red plates, 2 yellow plates, 1 green plate, and 3 blue plates. How many
ways can she stack her plates in a single tower? Show your work.
33. A conveyor belt sushi restaurant lets you choose what to eat from different-coloured plates that go
by. You pay based on the colours of the plates.
After lunch, Ming stacks her 2 red plates and 4 green plates while Rudo stacks his 3 green plates, 1
red plate, and 2 blue plates. Use the Fundamental Counting Principle to count how many ways
they can make the two stacks of plates. Show your work.
34. A conveyor belt sushi restaurant lets you choose what to eat from variously-coloured plates that go
by. You pay based on the colours of the plates.
After dinner, Claire stacks her 3 blue plates and 5 green plates while Bradley stacks his 3 red
plates, 1 green plate, 4 yellow plates, and 2 blue plates. Use the Fundamental Counting Principle to
count how many ways they can make the two stacks of plates. Show your work.
35. Two friends are building stacks of 12 coins. Stack 1 has 5 identical pennies, 3 identical nickels,
and 4 identical quarters. Stack 2 has 3 identical pennies, 3 identical nickels, and 6 identical
quarters.
Which set of coins can make more stacks of 12 coins? Show your work.
36. Two friends are building stacks of 15 coins. Stack 1 has 10 identical pennies, 3 identical nickels,
and 2 identical quarters. Stack 2 has 5 identical pennies, 2 identical nickels, and 8 identical
quarters.
Which set of coins can make more stacks of 12 coins? Show your work.
37. Kathy stacks 13 coins: 3 identical pennies, 4 identical nickels, 4 identical quarters, and 2 identical
dimes. How many different ways can Kathy stack the coins in a single tower in each situation
below. Show your work.
a) There are no conditions.
b) There must be a quarter on top and a quarter on the bottom.
38. Garrick stacks 15 coins: 9 identical pennies, 5 identical nickels, and 1 quarter. How many different
ways can Garrick stack the coins in a single tower in each situation below. Show your work.
a) There are no conditions.
b) There must be a penny on top.
39. Compare the number of different arrangements you can make using all the letters in the words
NANAIMO and FANNY BAY. Show your work.
40. Compare the number of different arrangements you can make using all the letters in the words
RED DEER and REGINA. Show your work.
41. From a group of eight students, four students need to be chosen for a dance committee. How many
committees are possible? Show your work.
42. From a group of 12 students, three students need to be chosen for an environmental committee.
How many committees are possible? Show your work.
43. From a group of seven students, four students need to be chosen for a graduation committee.
a) How many committees are possible? Show your work.
b) How many committees are possible, if only three students are needed on the committee?
c) Compare your answers for parts a) and b). What do you notice? Explain why this occurred.
44. A youth hostel has 3 rooms that contain 8, 5, and 3 beds, respectively. How many ways can the 16
players on a hockey team be assigned to these rooms? Show your work.
45. A youth hostel has 3 rooms that contain 6, 5, and 4 beds, respectively. How many ways can the 15
players on a hockey team be assigned to these rooms? Show your work.
46. There are 12 boys and 15 girls in an English classroom. A group of 5 students is needed to read
from a play. If there are 2 roles for boys, 2 roles for girls, and a narrator who could be a boy or a
girl, how many different groups of 5 students are possible? Show your work.
47. There are 18 boys and 13 girls in an English classroom. A group of 6 students is needed to read
from a play. If there are 2 roles for boys, 3 roles for girls, and a narrator who could be a boy or a
girl, how many different groups of 6 students are possible? Show your work.
48. There are 6 boys and 18 girls in a class. A group of 5 students is needed to work on a project. If at
least 2 boys are needed, how many different groups of 5 students are possible? Show your work.
49. a) Evaluate each of the following.
i)
ii)
iii)
iv)
b) What do you notice about the sums?
c) Use your answer to part a) to predict the value of the following:
50. Use what you know about combinations to determine which of the following values is the greatest,
without calculating. Explain your reasoning.
A.
B.
C.
D.
51. A hockey team is lining up in a row for a group photo. There team has 1 goalie, 4 defense, and 7
forwards. The photographer wants the defense on one side of the goalie and the forwards on the
other side. How many ways can the team stand in a row for this pose? Show your work.
52. A hockey team is lining up in a row for a group photo. The team has 2 goalies, 4 defense, 10
forwards, and a coach. The photographer wants the team grouped by position, with the goalies on
one end and the coach on the other. How many ways can the team stand in a row for this pose?
Show your work.
53. A hockey team is preparing for a group photo. The team has 1 goalie, 4 defense, and 7 forwards.
The photographer wants two rows of six players. How many ways can the team arrange six players
in the front row with at most 2 defense? Show your work.
54. A hockey team is preparing for a group photo. The team has 2 goalie, 6 defense, and 8 forwards.
The photographer wants two rows of eight players. How many ways can the team arrange eight
players in the front row with at least one goalie? Show your work.
55. Twelve camp counselors are signing up for training courses that have only a limited number of
spaces. Only 5 people can take the water safety course, 3 people can take the first aid course, 2
people can take the conflict management course, and 2 people can take the astronomy course. How
many ways can the 12 counselors be placed in the four courses? Show your work.
56. Fifteen camp counselors are signing up for training courses that have only a limited number of
spaces. Only 5 people can take the water safety course, 4 people can take the first aid course, 3
people can take the conflict management course, and 3 people can take the astronomy course. How
many ways can the 15 counselors be placed in the four courses? Show your work.
57. Three vehicles are taking a choir of 20 students to a recital. A minibus can take 12 students, an
SUV can take 5 students, and the remaining 3 students can ride with the choirmaster. How many
ways can the 20 students be assigned to the 3 vehicles? Show your work.
58. Three vehicles are taking a choir of 25 students to a recital. A minibus can take 16 students, an
SUV can take 6 students, and the remaining 3 students can ride with the choirmaster. How many
ways can the 20 students be assigned to the 3 vehicles? Show your work.
59. How many different five-card hands that contain at most two face cards (jack, queen, or king) can
be dealt to one person from a standard deck of playing cards? Show your work.
60. How many different four-card hands that contain at least two face cards (jack, queen, or king) can
be dealt to one person from a standard deck of playing cards? Show your work.
40s applied perm comb assign 2012
Answer Section
MULTIPLE CHOICE
1. ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 4.1
OBJ: 4.1 Represent and solve counting problems, using a graphic organizer. | 4.2 Generalize the
fundamental counting principle, using inductive reasoning. | 4.3 Identify and explain assumptions
made in solving a counting problem. | 4.4 Solve a contextual counting problem, using the
fundamental counting principle, and explain the reasoning.
TOP: Counting Principles
KEY: counting | Fundamental Counting Principle
2. ANS: C
PTS: 1
DIF: Grade 12
REF: Lesson 4.1
OBJ: 4.1 Represent and solve counting problems, using a graphic organizer. | 4.2 Generalize the
fundamental counting principle, using inductive reasoning. | 4.3 Identify and explain assumptions
made in solving a counting problem. | 4.4 Solve a contextual counting problem, using the
fundamental counting principle, and explain the reasoning.
TOP: Counting Principles
KEY: counting | Fundamental Counting Principle
3. ANS: C
PTS: 1
DIF: Grade 12
REF: Lesson 4.1
OBJ: 4.1 Represent and solve counting problems, using a graphic organizer. | 4.2 Generalize the
fundamental counting principle, using inductive reasoning. | 4.3 Identify and explain assumptions
made in solving a counting problem. | 4.4 Solve a contextual counting problem, using the
fundamental counting principle, and explain the reasoning.
TOP: Counting Principles
KEY: counting | Fundamental Counting Principle
4. ANS: D
PTS: 1
DIF: Grade 12
REF: Lesson 4.1
OBJ: 4.1 Represent and solve counting problems, using a graphic organizer. | 4.2 Generalize the
fundamental counting principle, using inductive reasoning. | 4.3 Identify and explain assumptions
made in solving a counting problem. | 4.4 Solve a contextual counting problem, using the
fundamental counting principle, and explain the reasoning.
TOP: Counting Principles
KEY: counting | Fundamental Counting Principle
5. ANS: A
PTS: 1
DIF: Grade 12
REF: Lesson 4.1
OBJ: 4.1 Represent and solve counting problems, using a graphic organizer. | 4.2 Generalize the
fundamental counting principle, using inductive reasoning. | 4.3 Identify and explain assumptions
made in solving a counting problem. | 4.4 Solve a contextual counting problem, using the
fundamental counting principle, and explain the reasoning.
TOP: Counting Principles
KEY: counting | Fundamental Counting Principle
6. ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 4.1
OBJ: 4.1 Represent and solve counting problems, using a graphic organizer. | 4.2 Generalize the
fundamental counting principle, using inductive reasoning. | 4.3 Identify and explain assumptions
made in solving a counting problem. | 4.4 Solve a contextual counting problem, using the
fundamental counting principle, and explain the reasoning.
TOP: Counting Principles
KEY: counting | Fundamental Counting Principle
7. ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 4.1
OBJ: 4.1 Represent and solve counting problems, using a graphic organizer. | 4.2 Generalize the
fundamental counting principle, using inductive reasoning. | 4.3 Identify and explain assumptions
8.
9.
10.
11.
12.
13.
14.
15.
made in solving a counting problem. | 4.4 Solve a contextual counting problem, using the
fundamental counting principle, and explain the reasoning.
TOP: Counting Principles
KEY: counting | Fundamental Counting Principle
ANS: C
PTS: 1
DIF: Grade 12
REF: Lesson 4.1
OBJ: 4.1 Represent and solve counting problems, using a graphic organizer. | 4.2 Generalize the
fundamental counting principle, using inductive reasoning. | 4.3 Identify and explain assumptions
made in solving a counting problem. | 4.4 Solve a contextual counting problem, using the
fundamental counting principle, and explain the reasoning.
TOP: Counting Principles
KEY: counting | Fundamental Counting Principle
ANS: C
PTS: 1
DIF: Grade 12
REF: Lesson 4.1
OBJ: 4.1 Represent and solve counting problems, using a graphic organizer. | 4.2 Generalize the
fundamental counting principle, using inductive reasoning. | 4.3 Identify and explain assumptions
made in solving a counting problem. | 4.4 Solve a contextual counting problem, using the
fundamental counting principle, and explain the reasoning.
TOP: Counting Principles
KEY: counting | Fundamental Counting Principle
ANS: A
PTS: 1
DIF: Grade 12
REF: Lesson 4.1
OBJ: 4.1 Represent and solve counting problems, using a graphic organizer. | 4.2 Generalize the
fundamental counting principle, using inductive reasoning. | 4.3 Identify and explain assumptions
made in solving a counting problem. | 4.4 Solve a contextual counting problem, using the
fundamental counting principle, and explain the reasoning.
TOP: Counting Principles
KEY: counting | Fundamental Counting Principle
ANS: D
PTS: 1
DIF: Grade 12
REF: Lesson 4.1
OBJ: 4.1 Represent and solve counting problems, using a graphic organizer. | 4.2 Generalize the
fundamental counting principle, using inductive reasoning. | 4.3 Identify and explain assumptions
made in solving a counting problem. | 4.4 Solve a contextual counting problem, using the
fundamental counting principle, and explain the reasoning.
TOP: Counting Principles
KEY: counting | Fundamental Counting Principle
ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 4.1
OBJ: 4.1 Represent and solve counting problems, using a graphic organizer. | 4.2 Generalize the
fundamental counting principle, using inductive reasoning. | 4.3 Identify and explain assumptions
made in solving a counting problem. | 4.4 Solve a contextual counting problem, using the
fundamental counting principle, and explain the reasoning.
TOP: Counting Principles
KEY: counting | Fundamental Counting Principle
ANS: D
PTS: 1
DIF: Grade 12
REF: Lesson 4.1
OBJ: 4.1 Represent and solve counting problems, using a graphic organizer. | 4.2 Generalize the
fundamental counting principle, using inductive reasoning. | 4.3 Identify and explain assumptions
made in solving a counting problem. | 4.4 Solve a contextual counting problem, using the
fundamental counting principle, and explain the reasoning.
TOP: Counting Principles
KEY: counting
ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 4.1
OBJ: 4.1 Represent and solve counting problems, using a graphic organizer. | 4.2 Generalize the
fundamental counting principle, using inductive reasoning. | 4.3 Identify and explain assumptions
made in solving a counting problem. | 4.4 Solve a contextual counting problem, using the
fundamental counting principle, and explain the reasoning.
TOP: Counting Principles
KEY: counting
ANS: D
PTS: 1
DIF: Grade 12
REF: Lesson 4.1
16.
17.
18.
19.
20.
21.
OBJ: 4.1 Represent and solve counting problems, using a graphic organizer. | 4.2 Generalize the
fundamental counting principle, using inductive reasoning. | 4.3 Identify and explain assumptions
made in solving a counting problem. | 4.4 Solve a contextual counting problem, using the
fundamental counting principle, and explain the reasoning.
TOP: Counting Principles
KEY: counting
ANS: A
PTS: 1
DIF: Grade 12
REF: Lesson 4.2
OBJ: 5.1 Represent the number of arrangements of n elements taken n at a time, using factorial
notation. | 5.2 Determine, with or without technology, the value of a factorial. | 5.3 Simplify a
numeric or algebraic fraction containing factorials in both the numerator and denominator. | 5.4
Solve an equation that involves factorials.
TOP: Introducing Permutations and Factorial Notation
KEY: permutation | factorial notation
ANS: D
PTS: 1
DIF: Grade 12
REF: Lesson 4.2
OBJ: 5.1 Represent the number of arrangements of n elements taken n at a time, using factorial
notation. | 5.2 Determine, with or without technology, the value of a factorial. | 5.3 Simplify a
numeric or algebraic fraction containing factorials in both the numerator and denominator. | 5.4
Solve an equation that involves factorials.
TOP: Introducing Permutations and Factorial Notation
KEY: permutation | factorial notation
ANS: C
PTS: 1
DIF: Grade 12
REF: Lesson 4.2
OBJ: 5.1 Represent the number of arrangements of n elements taken n at a time, using factorial
notation. | 5.2 Determine, with or without technology, the value of a factorial. | 5.3 Simplify a
numeric or algebraic fraction containing factorials in both the numerator and denominator. | 5.4
Solve an equation that involves factorials.
TOP: Introducing Permutations and Factorial Notation
KEY: permutation | factorial notation
ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 4.2
OBJ: 5.1 Represent the number of arrangements of n elements taken n at a time, using factorial
notation. | 5.2 Determine, with or without technology, the value of a factorial. | 5.3 Simplify a
numeric or algebraic fraction containing factorials in both the numerator and denominator. | 5.4
Solve an equation that involves factorials.
TOP: Introducing Permutations and Factorial Notation
KEY: permutation | factorial notation
ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 4.2
OBJ: 5.1 Represent the number of arrangements of n elements taken n at a time, using factorial
notation. | 5.2 Determine, with or without technology, the value of a factorial. | 5.3 Simplify a
numeric or algebraic fraction containing factorials in both the numerator and denominator. | 5.4
Solve an equation that involves factorials.
TOP: Introducing Permutations and Factorial Notation
KEY: permutation | factorial notation
ANS: C
PTS: 1
DIF: Grade 12
REF: Lesson 4.2
OBJ: 5.1 Represent the number of arrangements of n elements taken n at a time, using factorial
notation. | 5.2 Determine, with or without technology, the value of a factorial. | 5.3 Simplify a
numeric or algebraic fraction containing factorials in both the numerator and denominator. | 5.4
Solve an equation that involves factorials.
TOP: Introducing Permutations and Factorial Notation
22.
23.
24.
25.
26.
27.
28.
KEY: permutation | factorial notation
ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 4.2
OBJ: 5.1 Represent the number of arrangements of n elements taken n at a time, using factorial
notation. | 5.2 Determine, with or without technology, the value of a factorial. | 5.3 Simplify a
numeric or algebraic fraction containing factorials in both the numerator and denominator. | 5.4
Solve an equation that involves factorials.
TOP: Introducing Permutations and Factorial Notation
KEY: permutation | factorial notation
ANS: A
PTS: 1
DIF: Grade 12
REF: Lesson 4.2
OBJ: 5.1 Represent the number of arrangements of n elements taken n at a time, using factorial
notation. | 5.2 Determine, with or without technology, the value of a factorial. | 5.3 Simplify a
numeric or algebraic fraction containing factorials in both the numerator and denominator. | 5.4
Solve an equation that involves factorials.
TOP: Introducing Permutations and Factorial Notation
KEY: permutation | factorial notation
ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 4.2
OBJ: 5.1 Represent the number of arrangements of n elements taken n at a time, using factorial
notation. | 5.2 Determine, with or without technology, the value of a factorial. | 5.3 Simplify a
numeric or algebraic fraction containing factorials in both the numerator and denominator. | 5.4
Solve an equation that involves factorials.
TOP: Introducing Permutations and Factorial Notation
KEY: permutation | factorial notation
ANS: C
PTS: 1
DIF: Grade 12
REF: Lesson 4.2
OBJ: 5.1 Represent the number of arrangements of n elements taken n at a time, using factorial
notation. | 5.2 Determine, with or without technology, the value of a factorial. | 5.3 Simplify a
numeric or algebraic fraction containing factorials in both the numerator and denominator. | 5.4
Solve an equation that involves factorials.
TOP: Introducing Permutations and Factorial Notation
KEY: permutation | factorial notation
ANS: C
PTS: 1
DIF: Grade 12
REF: Lesson 4.2
OBJ: 5.1 Represent the number of arrangements of n elements taken n at a time, using factorial
notation. | 5.2 Determine, with or without technology, the value of a factorial. | 5.3 Simplify a
numeric or algebraic fraction containing factorials in both the numerator and denominator. | 5.4
Solve an equation that involves factorials.
TOP: Introducing Permutations and Factorial Notation
KEY: permutation | factorial notation
ANS: A
PTS: 1
DIF: Grade 12
REF: Lesson 4.2
OBJ: 5.1 Represent the number of arrangements of n elements taken n at a time, using factorial
notation. | 5.2 Determine, with or without technology, the value of a factorial. | 5.3 Simplify a
numeric or algebraic fraction containing factorials in both the numerator and denominator. | 5.4
Solve an equation that involves factorials.
TOP: Introducing Permutations and Factorial Notation
KEY: permutation | factorial notation
ANS: C
PTS: 1
DIF: Grade 12
REF: Lesson 4.2
OBJ: 5.1 Represent the number of arrangements of n elements taken n at a time, using factorial
notation. | 5.2 Determine, with or without technology, the value of a factorial. | 5.3 Simplify a
29.
30.
31.
32.
33.
34.
35.
36.
37.
numeric or algebraic fraction containing factorials in both the numerator and denominator. | 5.4
Solve an equation that involves factorials.
TOP: Introducing Permutations and Factorial Notation
KEY: permutation | factorial notation
ANS: D
PTS: 1
DIF: Grade 12
REF: Lesson 4.2
OBJ: 5.1 Represent the number of arrangements of n elements taken n at a time, using factorial
notation. | 5.2 Determine, with or without technology, the value of a factorial. | 5.3 Simplify a
numeric or algebraic fraction containing factorials in both the numerator and denominator. | 5.4
Solve an equation that involves factorials.
TOP: Introducing Permutations and Factorial Notation
KEY: permutation | factorial notation
ANS: D
PTS: 1
DIF: Grade 12
REF: Lesson 4.2
OBJ: 5.1 Represent the number of arrangements of n elements taken n at a time, using factorial
notation. | 5.2 Determine, with or without technology, the value of a factorial. | 5.3 Simplify a
numeric or algebraic fraction containing factorials in both the numerator and denominator. | 5.4
Solve an equation that involves factorials.
TOP: Introducing Permutations and Factorial Notation
KEY: permutation | factorial notation
ANS: C
PTS: 1
DIF: Grade 12
REF: Lesson 4.3
OBJ: 5.5 Determine the number of permutations of n elements taken r at a time. | 5.8 Generalize
strategies for determining the number of permutations of n elements taken r at a time.
TOP: Permutations When All Objects Are Distinguishable
KEY: permutation
ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 4.3
OBJ: 5.5 Determine the number of permutations of n elements taken r at a time. | 5.8 Generalize
strategies for determining the number of permutations of n elements taken r at a time.
TOP: Permutations When All Objects Are Distinguishable
KEY: permutation
ANS: A
PTS: 1
DIF: Grade 12
REF: Lesson 4.3
OBJ: 5.5 Determine the number of permutations of n elements taken r at a time. | 5.8 Generalize
strategies for determining the number of permutations of n elements taken r at a time.
TOP: Permutations When All Objects Are Distinguishable
KEY: permutation
ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 4.3
OBJ: 5.5 Determine the number of permutations of n elements taken r at a time. | 5.8 Generalize
strategies for determining the number of permutations of n elements taken r at a time.
TOP: Permutations When All Objects Are Distinguishable
KEY: permutation
ANS: D
PTS: 1
DIF: Grade 12
REF: Lesson 4.3
OBJ: 5.5 Determine the number of permutations of n elements taken r at a time. | 5.8 Generalize
strategies for determining the number of permutations of n elements taken r at a time.
TOP: Permutations When All Objects Are Distinguishable
KEY: permutation
ANS: C
PTS: 1
DIF: Grade 12
REF: Lesson 4.3
OBJ: 5.5 Determine the number of permutations of n elements taken r at a time. | 5.8 Generalize
strategies for determining the number of permutations of n elements taken r at a time.
TOP: Permutations When All Objects Are Distinguishable
KEY: permutation
ANS: A
PTS: 1
DIF: Grade 12
REF: Lesson 4.3
OBJ: 5.5 Determine the number of permutations of n elements taken r at a time. | 5.8 Generalize
strategies for determining the number of permutations of n elements taken r at a time.
TOP: Permutations When All Objects Are Distinguishable
KEY: permutation
38. ANS: A
PTS: 1
DIF: Grade 12
REF: Lesson 4.3
OBJ: 5.5 Determine the number of permutations of n elements taken r at a time. | 5.8 Generalize
strategies for determining the number of permutations of n elements taken r at a time.
TOP: Permutations When All Objects Are Distinguishable
KEY: permutation
39. ANS: D
PTS: 1
DIF: Grade 12
REF: Lesson 4.3
OBJ: 5.5 Determine the number of permutations of n elements taken r at a time. | 5.8 Generalize
strategies for determining the number of permutations of n elements taken r at a time.
TOP: Permutations When All Objects Are Distinguishable
KEY: permutation
40. ANS: A
PTS: 1
DIF: Grade 12
REF: Lesson 4.3
OBJ: 5.5 Determine the number of permutations of n elements taken r at a time. | 5.8 Generalize
strategies for determining the number of permutations of n elements taken r at a time.
TOP: Permutations When All Objects Are Distinguishable
KEY: permutation
41. ANS: D
PTS: 1
DIF: Grade 12
REF: Lesson 4.3
OBJ: 5.5 Determine the number of permutations of n elements taken r at a time. | 5.8 Generalize
strategies for determining the number of permutations of n elements taken r at a time.
TOP: Permutations When All Objects Are Distinguishable
KEY: permutation
42. ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 4.3
OBJ: 5.5 Determine the number of permutations of n elements taken r at a time. | 5.8 Generalize
strategies for determining the number of permutations of n elements taken r at a time.
TOP: Permutations When All Objects Are Distinguishable
KEY: permutation
43. ANS: A
PTS: 1
DIF: Grade 12
REF: Lesson 4.3
OBJ: 5.5 Determine the number of permutations of n elements taken r at a time. | 5.8 Generalize
strategies for determining the number of permutations of n elements taken r at a time.
TOP: Permutations When All Objects Are Distinguishable
KEY: permutation
44. ANS: A
PTS: 1
DIF: Grade 12
REF: Lesson 4.3
OBJ: 5.5 Determine the number of permutations of n elements taken r at a time. | 5.8 Generalize
strategies for determining the number of permutations of n elements taken r at a time.
TOP: Permutations When All Objects Are Distinguishable
KEY: permutation
45. ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 4.3
OBJ: 5.5 Determine the number of permutations of n elements taken r at a time. | 5.8 Generalize
strategies for determining the number of permutations of n elements taken r at a time.
TOP: Permutations When All Objects Are Distinguishable
KEY: permutation
46. ANS: D
PTS: 1
DIF: Grade 12
REF: Lesson 4.4
OBJ: 5.6 Determine the number of permutations of n elements taken n at a time where some
elements are not distinct. | 5.7 Explain, using examples, the effect on the total number of
permutations of n elements when two or more elements are identical.
TOP: Permutations When Objects Are Identical
KEY: permutation | factorial notation
47. ANS: D
PTS: 1
DIF: Grade 12
REF: Lesson 4.4
OBJ: 5.6 Determine the number of permutations of n elements taken n at a time where some
elements are not distinct. | 5.7 Explain, using examples, the effect on the total number of
permutations of n elements when two or more elements are identical.
TOP: Permutations When Objects Are Identical
KEY: permutation | factorial notation
48. ANS: A
PTS: 1
DIF: Grade 12
REF: Lesson 4.4
OBJ: 5.6 Determine the number of permutations of n elements taken n at a time where some
49.
50.
51.
52.
53.
54.
55.
elements are not distinct. | 5.7 Explain, using examples, the effect on the total number of
permutations of n elements when two or more elements are identical.
TOP: Permutations When Objects Are Identical
KEY: permutation | factorial notation
ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 4.4
OBJ: 5.6 Determine the number of permutations of n elements taken n at a time where some
elements are not distinct. | 5.7 Explain, using examples, the effect on the total number of
permutations of n elements when two or more elements are identical.
TOP: Permutations When Objects Are Identical
KEY: permutation | factorial notation
ANS: A
PTS: 1
DIF: Grade 12
REF: Lesson 4.4
OBJ: 5.6 Determine the number of permutations of n elements taken n at a time where some
elements are not distinct. | 5.7 Explain, using examples, the effect on the total number of
permutations of n elements when two or more elements are identical.
TOP: Permutations When Objects Are Identical
KEY: permutation | factorial notation
ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 4.4
OBJ: 5.6 Determine the number of permutations of n elements taken n at a time where some
elements are not distinct. | 5.7 Explain, using examples, the effect on the total number of
permutations of n elements when two or more elements are identical.
TOP: Permutations When Objects Are Identical
KEY: permutation | factorial notation
ANS: C
PTS: 1
DIF: Grade 12
REF: Lesson 4.4
OBJ: 5.6 Determine the number of permutations of n elements taken n at a time where some
elements are not distinct. | 5.7 Explain, using examples, the effect on the total number of
permutations of n elements when two or more elements are identical.
TOP: Permutations When Objects Are Identical
KEY: permutation | factorial notation
ANS: A
PTS: 1
DIF: Grade 12
REF: Lesson 4.4
OBJ: 5.6 Determine the number of permutations of n elements taken n at a time where some
elements are not distinct. | 5.7 Explain, using examples, the effect on the total number of
permutations of n elements when two or more elements are identical.
TOP: Permutations When Objects Are Identical
KEY: permutation | factorial notation
ANS: C
PTS: 1
DIF: Grade 12
REF: Lesson 4.4
OBJ: 5.6 Determine the number of permutations of n elements taken n at a time where some
elements are not distinct. | 5.7 Explain, using examples, the effect on the total number of
permutations of n elements when two or more elements are identical.
TOP: Permutations When Objects Are Identical
KEY: permutation | factorial notation
ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 4.4
OBJ: 5.6 Determine the number of permutations of n elements taken n at a time where some
elements are not distinct. | 5.7 Explain, using examples, the effect on the total number of
permutations of n elements when two or more elements are identical.
TOP: Permutations When Objects Are Identical
KEY: permutation | factorial notation
56. ANS: D
PTS: 1
DIF: Grade 12
REF: Lesson 4.4
OBJ: 5.6 Determine the number of permutations of n elements taken n at a time where some
elements are not distinct. | 5.7 Explain, using examples, the effect on the total number of
permutations of n elements when two or more elements are identical.
TOP: Permutations When Objects Are Identical
KEY: permutation | factorial notation
57. ANS: C
PTS: 1
DIF: Grade 12
REF: Lesson 4.4
OBJ: 5.6 Determine the number of permutations of n elements taken n at a time where some
elements are not distinct. | 5.7 Explain, using examples, the effect on the total number of
permutations of n elements when two or more elements are identical.
TOP: Permutations When Objects Are Identical
KEY: permutation | factorial notation
58. ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 4.4
OBJ: 5.6 Determine the number of permutations of n elements taken n at a time where some
elements are not distinct. | 5.7 Explain, using examples, the effect on the total number of
permutations of n elements when two or more elements are identical.
TOP: Permutations When Objects Are Identical
KEY: permutation | factorial notation
59. ANS: D
PTS: 1
DIF: Grade 12
REF: Lesson 4.4
OBJ: 5.6 Determine the number of permutations of n elements taken n at a time where some
elements are not distinct. | 5.7 Explain, using examples, the effect on the total number of
permutations of n elements when two or more elements are identical.
TOP: Permutations When Objects Are Identical
KEY: permutation | factorial notation
60. ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 4.4
OBJ: 5.6 Determine the number of permutations of n elements taken n at a time where some
elements are not distinct. | 5.7 Explain, using examples, the effect on the total number of
permutations of n elements when two or more elements are identical.
TOP: Permutations When Objects Are Identical
KEY: permutation | factorial notation
61. ANS: A
PTS: 1
DIF: Grade 12
REF: Lesson 4.5
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 5.1 Represent
the number of arrangements of n elements taken n at a time, using factorial notation.
TOP: Exploring Combinations
KEY: counting | combination | factorial notation
62. ANS: C
PTS: 1
DIF: Grade 12
REF: Lesson 4.5
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 5.1 Represent
the number of arrangements of n elements taken n at a time, using factorial notation.
TOP: Exploring Combinations
KEY: counting | combination | factorial notation
63. ANS: C
PTS: 1
DIF: Grade 12
REF: Lesson 4.5
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 5.1 Represent
the number of arrangements of n elements taken n at a time, using factorial notation.
TOP: Exploring Combinations
KEY: counting | combination | factorial notation
64. ANS: A
PTS: 1
DIF: Grade 12
REF: Lesson 4.5
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 5.1 Represent
the number of arrangements of n elements taken n at a time, using factorial notation.
TOP: Exploring Combinations
KEY: counting | combination | factorial notation
65. ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 4.5
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 5.1 Represent
the number of arrangements of n elements taken n at a time, using factorial notation.
TOP: Exploring Combinations
KEY: counting | combination | factorial notation
66. ANS: C
PTS: 1
DIF: Grade 12
REF: Lesson 4.6
OBJ: 6.1 Explain, using examples, why order is or is not important when solving problems that
involve permutations or combinations. | 6.2 Determine the number of combinations of n elements
taken r at a time. | 6.3 Generalize strategies for determining the number of combinations of n
elements taken r at a time.
TOP: Combinations
KEY: counting | combination
67. ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 4.6
OBJ: 6.1 Explain, using examples, why order is or is not important when solving problems that
involve permutations or combinations. | 6.2 Determine the number of combinations of n elements
taken r at a time. | 6.3 Generalize strategies for determining the number of combinations of n
elements taken r at a time.
TOP: Combinations
KEY: counting | combination
68. ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 4.6
OBJ: 6.1 Explain, using examples, why order is or is not important when solving problems that
involve permutations or combinations. | 6.2 Determine the number of combinations of n elements
taken r at a time. | 6.3 Generalize strategies for determining the number of combinations of n
elements taken r at a time.
TOP: Combinations
KEY: counting | combination
69. ANS: A
PTS: 1
DIF: Grade 12
REF: Lesson 4.6
OBJ: 6.1 Explain, using examples, why order is or is not important when solving problems that
involve permutations or combinations. | 6.2 Determine the number of combinations of n elements
taken r at a time. | 6.3 Generalize strategies for determining the number of combinations of n
elements taken r at a time.
TOP: Combinations
KEY: counting | combination
70. ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 4.6
OBJ: 6.1 Explain, using examples, why order is or is not important when solving problems that
involve permutations or combinations. | 6.2 Determine the number of combinations of n elements
taken r at a time. | 6.3 Generalize strategies for determining the number of combinations of n
elements taken r at a time.
TOP: Combinations
KEY: counting | combination
71. ANS: A
PTS: 1
DIF: Grade 12
REF: Lesson 4.6
OBJ: 6.1 Explain, using examples, why order is or is not important when solving problems that
involve permutations or combinations. | 6.2 Determine the number of combinations of n elements
taken r at a time. | 6.3 Generalize strategies for determining the number of combinations of n
elements taken r at a time.
TOP: Combinations
KEY: counting | combination
72. ANS: D
PTS: 1
DIF: Grade 12
REF: Lesson 4.6
OBJ: 6.1 Explain, using examples, why order is or is not important when solving problems that
involve permutations or combinations. | 6.2 Determine the number of combinations of n elements
taken r at a time. | 6.3 Generalize strategies for determining the number of combinations of n
elements taken r at a time.
TOP: Combinations
KEY: counting | combination
73. ANS: A
PTS: 1
DIF: Grade 12
REF: Lesson 4.6
OBJ: 6.1 Explain, using examples, why order is or is not important when solving problems that
involve permutations or combinations. | 6.2 Determine the number of combinations of n elements
taken r at a time. | 6.3 Generalize strategies for determining the number of combinations of n
elements taken r at a time.
TOP: Combinations
KEY: counting | combination
74. ANS: C
PTS: 1
DIF: Grade 12
REF: Lesson 4.6
OBJ: 6.1 Explain, using examples, why order is or is not important when solving problems that
involve permutations or combinations. | 6.2 Determine the number of combinations of n elements
taken r at a time. | 6.3 Generalize strategies for determining the number of combinations of n
elements taken r at a time.
TOP: Combinations
KEY: counting | combination
75. ANS: D
PTS: 1
DIF: Grade 12
REF: Lesson 4.6
OBJ: 6.1 Explain, using examples, why order is or is not important when solving problems that
involve permutations or combinations. | 6.2 Determine the number of combinations of n elements
taken r at a time. | 6.3 Generalize strategies for determining the number of combinations of n
elements taken r at a time.
TOP: Combinations
KEY: counting | combination
76. ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 4.6
OBJ: 6.1 Explain, using examples, why order is or is not important when solving problems that
involve permutations or combinations. | 6.2 Determine the number of combinations of n elements
taken r at a time. | 6.3 Generalize strategies for determining the number of combinations of n
elements taken r at a time.
TOP: Combinations
KEY: counting | combination
77. ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 4.6
OBJ: 6.1 Explain, using examples, why order is or is not important when solving problems that
involve permutations or combinations. | 6.2 Determine the number of combinations of n elements
taken r at a time. | 6.3 Generalize strategies for determining the number of combinations of n
elements taken r at a time.
TOP: Combinations
KEY: counting | combination
78. ANS: C
PTS: 1
DIF: Grade 12
REF: Lesson 4.6
OBJ: 6.1 Explain, using examples, why order is or is not important when solving problems that
involve permutations or combinations. | 6.2 Determine the number of combinations of n elements
taken r at a time. | 6.3 Generalize strategies for determining the number of combinations of n
elements taken r at a time.
TOP: Combinations
KEY: counting | combination
79. ANS: C
PTS: 1
DIF: Grade 12
REF: Lesson 4.6
OBJ: 6.1 Explain, using examples, why order is or is not important when solving problems that
involve permutations or combinations. | 6.2 Determine the number of combinations of n elements
taken r at a time. | 6.3 Generalize strategies for determining the number of combinations of n
elements taken r at a time.
TOP: Combinations
KEY: counting | combination
80. ANS: D
PTS: 1
DIF: Grade 12
REF: Lesson 4.6
OBJ: 6.1 Explain, using examples, why order is or is not important when solving problems that
involve permutations or combinations. | 6.2 Determine the number of combinations of n elements
taken r at a time. | 6.3 Generalize strategies for determining the number of combinations of n
81.
82.
83.
84.
85.
86.
87.
88.
elements taken r at a time.
TOP: Combinations
KEY: counting | combination
ANS: C
PTS: 1
DIF: Grade 12
REF: Lesson 4.7
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 4.4 Solve a
contextual counting problem, using the fundamental counting principle, and explain the reasoning.
| 5.5 Determine the number of permutations of n elements taken r at a time. | 6.2 Determine the
number of combinations of n elements taken r at a time.
TOP: Solving Counting Problems
KEY: counting | permutation | combination
ANS: D
PTS: 1
DIF: Grade 12
REF: Lesson 4.7
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 4.4 Solve a
contextual counting problem, using the fundamental counting principle, and explain the reasoning.
| 5.5 Determine the number of permutations of n elements taken r at a time. | 6.2 Determine the
number of combinations of n elements taken r at a time.
TOP: Solving Counting Problems
KEY: counting | permutation | combination
ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 4.7
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 4.4 Solve a
contextual counting problem, using the fundamental counting principle, and explain the reasoning.
| 5.5 Determine the number of permutations of n elements taken r at a time. | 6.2 Determine the
number of combinations of n elements taken r at a time.
TOP: Solving Counting Problems
KEY: counting | permutation | combination
ANS: A
PTS: 1
DIF: Grade 12
REF: Lesson 4.7
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 4.4 Solve a
contextual counting problem, using the fundamental counting principle, and explain the reasoning.
| 5.5 Determine the number of permutations of n elements taken r at a time. | 6.2 Determine the
number of combinations of n elements taken r at a time.
TOP: Solving Counting Problems
KEY: counting | permutation | combination
ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 4.7
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 4.4 Solve a
contextual counting problem, using the fundamental counting principle, and explain the reasoning.
| 5.5 Determine the number of permutations of n elements taken r at a time. | 6.2 Determine the
number of combinations of n elements taken r at a time.
TOP: Solving Counting Problems
KEY: counting | permutation | combination
ANS: C
PTS: 1
DIF: Grade 12
REF: Lesson 4.7
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 4.4 Solve a
contextual counting problem, using the fundamental counting principle, and explain the reasoning.
| 5.5 Determine the number of permutations of n elements taken r at a time. | 6.2 Determine the
number of combinations of n elements taken r at a time.
TOP: Solving Counting Problems
KEY: counting | permutation | combination
ANS: A
PTS: 1
DIF: Grade 12
REF: Lesson 4.7
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 4.4 Solve a
contextual counting problem, using the fundamental counting principle, and explain the reasoning.
| 5.5 Determine the number of permutations of n elements taken r at a time. | 6.2 Determine the
number of combinations of n elements taken r at a time.
TOP: Solving Counting Problems
KEY: counting | permutation | combination
ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 4.7
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 4.4 Solve a
89.
90.
91.
92.
93.
94.
95.
contextual counting problem, using the fundamental counting principle, and explain the reasoning.
| 5.5 Determine the number of permutations of n elements taken r at a time. | 6.2 Determine the
number of combinations of n elements taken r at a time.
TOP: Solving Counting Problems
KEY: counting | permutation | combination
ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 4.7
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 4.4 Solve a
contextual counting problem, using the fundamental counting principle, and explain the reasoning.
| 5.5 Determine the number of permutations of n elements taken r at a time. | 6.2 Determine the
number of combinations of n elements taken r at a time.
TOP: Solving Counting Problems
KEY: counting | permutation
ANS: B
PTS: 1
DIF: Grade 12
REF: Lesson 4.7
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 4.4 Solve a
contextual counting problem, using the fundamental counting principle, and explain the reasoning.
| 5.5 Determine the number of permutations of n elements taken r at a time. | 6.2 Determine the
number of combinations of n elements taken r at a time.
TOP: Solving Counting Problems
KEY: counting | permutation
ANS: A
PTS: 1
DIF: Grade 12
REF: Lesson 4.7
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 4.4 Solve a
contextual counting problem, using the fundamental counting principle, and explain the reasoning.
| 5.5 Determine the number of permutations of n elements taken r at a time. | 6.2 Determine the
number of combinations of n elements taken r at a time.
TOP: Solving Counting Problems
KEY: counting | permutation
ANS: D
PTS: 1
DIF: Grade 12
REF: Lesson 4.7
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 4.4 Solve a
contextual counting problem, using the fundamental counting principle, and explain the reasoning.
| 5.5 Determine the number of permutations of n elements taken r at a time. | 6.2 Determine the
number of combinations of n elements taken r at a time.
TOP: Solving Counting Problems
KEY: counting | combination | Fundamental Counting Principle
ANS: A
PTS: 1
DIF: Grade 12
REF: Lesson 4.7
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 4.4 Solve a
contextual counting problem, using the fundamental counting principle, and explain the reasoning.
| 5.5 Determine the number of permutations of n elements taken r at a time. | 6.2 Determine the
number of combinations of n elements taken r at a time.
TOP: Solving Counting Problems
KEY: counting | combination | Fundamental Counting Principle
ANS: C
PTS: 1
DIF: Grade 12
REF: Lesson 4.7
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 4.4 Solve a
contextual counting problem, using the fundamental counting principle, and explain the reasoning.
| 5.5 Determine the number of permutations of n elements taken r at a time. | 6.2 Determine the
number of combinations of n elements taken r at a time.
TOP: Solving Counting Problems
KEY: counting | combination | Fundamental Counting Principle
ANS: D
PTS: 1
DIF: Grade 12
REF: Lesson 4.7
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 4.4 Solve a
contextual counting problem, using the fundamental counting principle, and explain the reasoning.
| 5.5 Determine the number of permutations of n elements taken r at a time. | 6.2 Determine the
number of combinations of n elements taken r at a time.
TOP: Solving Counting Problems
KEY: counting | combination | Fundamental Counting Principle
SHORT ANSWER
1. ANS:
no
PTS: 1
DIF: Grade 12
REF: Lesson 4.1
OBJ: 4.1 Represent and solve counting problems, using a graphic organizer. | 4.2 Generalize the
fundamental counting principle, using inductive reasoning. | 4.3 Identify and explain assumptions
made in solving a counting problem. | 4.4 Solve a contextual counting problem, using the
fundamental counting principle, and explain the reasoning.
TOP: Counting Principles
KEY: counting | Fundamental Counting Principle
2. ANS:
yes
PTS: 1
DIF: Grade 12
REF: Lesson 4.1
OBJ: 4.1 Represent and solve counting problems, using a graphic organizer. | 4.2 Generalize the
fundamental counting principle, using inductive reasoning. | 4.3 Identify and explain assumptions
made in solving a counting problem. | 4.4 Solve a contextual counting problem, using the
fundamental counting principle, and explain the reasoning.
TOP: Counting Principles
KEY: counting | Fundamental Counting Principle
3. ANS:
yes
PTS: 1
DIF: Grade 12
REF: Lesson 4.1
OBJ: 4.1 Represent and solve counting problems, using a graphic organizer. | 4.2 Generalize the
fundamental counting principle, using inductive reasoning. | 4.3 Identify and explain assumptions
made in solving a counting problem. | 4.4 Solve a contextual counting problem, using the
fundamental counting principle, and explain the reasoning.
TOP: Counting Principles
KEY: counting | Fundamental Counting Principle
4. ANS:
12
PTS: 1
DIF: Grade 12
REF: Lesson 4.1
OBJ: 4.1 Represent and solve counting problems, using a graphic organizer. | 4.2 Generalize the
fundamental counting principle, using inductive reasoning. | 4.3 Identify and explain assumptions
made in solving a counting problem. | 4.4 Solve a contextual counting problem, using the
fundamental counting principle, and explain the reasoning.
TOP: Counting Principles
KEY: counting | Fundamental Counting Principle
5. ANS:
48
PTS: 1
DIF: Grade 12
REF: Lesson 4.1
OBJ: 4.1 Represent and solve counting problems, using a graphic organizer. | 4.2 Generalize the
fundamental counting principle, using inductive reasoning. | 4.3 Identify and explain assumptions
made in solving a counting problem. | 4.4 Solve a contextual counting problem, using the
fundamental counting principle, and explain the reasoning.
TOP: Counting Principles
KEY: counting | Fundamental Counting Principle
6. ANS:
200
PTS: 1
DIF: Grade 12
REF: Lesson 4.1
OBJ: 4.1 Represent and solve counting problems, using a graphic organizer. | 4.2 Generalize the
fundamental counting principle, using inductive reasoning. | 4.3 Identify and explain assumptions
made in solving a counting problem. | 4.4 Solve a contextual counting problem, using the
fundamental counting principle, and explain the reasoning.
TOP: Counting Principles
KEY: counting | Fundamental Counting Principle
7. ANS:
3375
PTS: 1
DIF: Grade 12
REF: Lesson 4.1
OBJ: 4.1 Represent and solve counting problems, using a graphic organizer. | 4.2 Generalize the
fundamental counting principle, using inductive reasoning. | 4.3 Identify and explain assumptions
made in solving a counting problem. | 4.4 Solve a contextual counting problem, using the
fundamental counting principle, and explain the reasoning.
TOP: Counting Principles
KEY: counting | Fundamental Counting Principle
8. ANS:
2730
PTS: 1
DIF: Grade 12
REF: Lesson 4.1
OBJ: 4.1 Represent and solve counting problems, using a graphic organizer. | 4.2 Generalize the
fundamental counting principle, using inductive reasoning. | 4.3 Identify and explain assumptions
made in solving a counting problem. | 4.4 Solve a contextual counting problem, using the
fundamental counting principle, and explain the reasoning.
TOP: Counting Principles
KEY: counting | Fundamental Counting Principle
9. ANS:
360
PTS: 1
DIF: Grade 12
REF: Lesson 4.1
OBJ: 4.1 Represent and solve counting problems, using a graphic organizer. | 4.2 Generalize the
fundamental counting principle, using inductive reasoning. | 4.3 Identify and explain assumptions
made in solving a counting problem. | 4.4 Solve a contextual counting problem, using the
fundamental counting principle, and explain the reasoning.
TOP: Counting Principles
KEY: counting | Fundamental Counting Principle
10. ANS:
1440
PTS:
1
DIF:
Grade 12
REF: Lesson 4.1
OBJ: 4.1 Represent and solve counting problems, using a graphic organizer. | 4.2 Generalize the
fundamental counting principle, using inductive reasoning. | 4.3 Identify and explain assumptions
made in solving a counting problem. | 4.4 Solve a contextual counting problem, using the
fundamental counting principle, and explain the reasoning.
TOP: Counting Principles
KEY: counting | Fundamental Counting Principle
11. ANS:
2640
PTS: 1
DIF: Grade 12
REF: Lesson 4.1
OBJ: 4.1 Represent and solve counting problems, using a graphic organizer. | 4.2 Generalize the
fundamental counting principle, using inductive reasoning. | 4.3 Identify and explain assumptions
made in solving a counting problem. | 4.4 Solve a contextual counting problem, using the
fundamental counting principle, and explain the reasoning.
TOP: Counting Principles
KEY: counting | Fundamental Counting Principle
12. ANS:
7
PTS: 1
DIF: Grade 12
REF: Lesson 4.1
OBJ: 4.1 Represent and solve counting problems, using a graphic organizer. | 4.2 Generalize the
fundamental counting principle, using inductive reasoning. | 4.3 Identify and explain assumptions
made in solving a counting problem. | 4.4 Solve a contextual counting problem, using the
fundamental counting principle, and explain the reasoning.
TOP: Counting Principles
KEY: counting | Fundamental Counting Principle
13. ANS:
8
PTS: 1
DIF: Grade 12
REF: Lesson 4.1
OBJ: 4.1 Represent and solve counting problems, using a graphic organizer. | 4.2 Generalize the
fundamental counting principle, using inductive reasoning. | 4.3 Identify and explain assumptions
made in solving a counting problem. | 4.4 Solve a contextual counting problem, using the
fundamental counting principle, and explain the reasoning.
TOP: Counting Principles
KEY: counting | Fundamental Counting Principle
14. ANS:
6
PTS: 1
DIF: Grade 12
REF: Lesson 4.1
OBJ: 4.1 Represent and solve counting problems, using a graphic organizer. | 4.2 Generalize the
fundamental counting principle, using inductive reasoning. | 4.3 Identify and explain assumptions
made in solving a counting problem. | 4.4 Solve a contextual counting problem, using the
fundamental counting principle, and explain the reasoning.
TOP: Counting Principles
KEY: counting | Fundamental Counting Principle
15. ANS:
11
PTS: 1
DIF: Grade 12
REF: Lesson 4.1
OBJ: 4.1 Represent and solve counting problems, using a graphic organizer. | 4.2 Generalize the
fundamental counting principle, using inductive reasoning. | 4.3 Identify and explain assumptions
made in solving a counting problem. | 4.4 Solve a contextual counting problem, using the
fundamental counting principle, and explain the reasoning.
TOP: Counting Principles
KEY: counting | Fundamental Counting Principle
16. ANS:
54
PTS: 1
DIF: Grade 12
REF: Lesson 4.2
OBJ: 5.1 Represent the number of arrangements of n elements taken n at a time, using factorial
notation. | 5.2 Determine, with or without technology, the value of a factorial. | 5.3 Simplify a
numeric or algebraic fraction containing factorials in both the numerator and denominator. | 5.4
Solve an equation that involves factorials.
TOP: Introducing Permutations and Factorial Notation
KEY: permutation | factorial notation
17. ANS:
11! or 39 916 800
PTS: 1
DIF: Grade 12
REF: Lesson 4.2
OBJ: 5.1 Represent the number of arrangements of n elements taken n at a time, using factorial
notation. | 5.2 Determine, with or without technology, the value of a factorial. | 5.3 Simplify a
numeric or algebraic fraction containing factorials in both the numerator and denominator. | 5.4
Solve an equation that involves factorials.
TOP: Introducing Permutations and Factorial Notation
KEY: permutation | factorial notation
18. ANS:
PTS: 1
DIF: Grade 12
REF: Lesson 4.2
OBJ: 5.1 Represent the number of arrangements of n elements taken n at a time, using factorial
notation. | 5.2 Determine, with or without technology, the value of a factorial. | 5.3 Simplify a
numeric or algebraic fraction containing factorials in both the numerator and denominator. | 5.4
Solve an equation that involves factorials.
TOP: Introducing Permutations and Factorial Notation
KEY: permutation | factorial notation
19. ANS:
288
PTS: 1
DIF: Grade 12
REF: Lesson 4.2
OBJ: 5.1 Represent the number of arrangements of n elements taken n at a time, using factorial
notation. | 5.2 Determine, with or without technology, the value of a factorial. | 5.3 Simplify a
numeric or algebraic fraction containing factorials in both the numerator and denominator. | 5.4
Solve an equation that involves factorials.
TOP: Introducing Permutations and Factorial Notation
KEY: permutation | factorial notation
20. ANS:
4
PTS: 1
DIF: Grade 12
REF: Lesson 4.2
OBJ: 5.1 Represent the number of arrangements of n elements taken n at a time, using factorial
notation. | 5.2 Determine, with or without technology, the value of a factorial. | 5.3 Simplify a
numeric or algebraic fraction containing factorials in both the numerator and denominator. | 5.4
Solve an equation that involves factorials.
TOP: Introducing Permutations and Factorial Notation
KEY: permutation | factorial notation
21. ANS:
7!
PTS: 1
DIF: Grade 12
REF: Lesson 4.2
OBJ: 5.1 Represent the number of arrangements of n elements taken n at a time, using factorial
notation. | 5.2 Determine, with or without technology, the value of a factorial. | 5.3 Simplify a
numeric or algebraic fraction containing factorials in both the numerator and denominator. | 5.4
Solve an equation that involves factorials.
TOP: Introducing Permutations and Factorial Notation
KEY: permutation | factorial notation
22. ANS:
PTS: 1
DIF: Grade 12
REF: Lesson 4.2
OBJ: 5.1 Represent the number of arrangements of n elements taken n at a time, using factorial
notation. | 5.2 Determine, with or without technology, the value of a factorial. | 5.3 Simplify a
numeric or algebraic fraction containing factorials in both the numerator and denominator. | 5.4
Solve an equation that involves factorials.
TOP: Introducing Permutations and Factorial Notation
KEY: permutation | factorial notation
23. ANS:
PTS: 1
DIF: Grade 12
REF: Lesson 4.2
OBJ: 5.1 Represent the number of arrangements of n elements taken n at a time, using factorial
notation. | 5.2 Determine, with or without technology, the value of a factorial. | 5.3 Simplify a
numeric or algebraic fraction containing factorials in both the numerator and denominator. | 5.4
Solve an equation that involves factorials.
TOP: Introducing Permutations and Factorial Notation
KEY: permutation | factorial notation
24. ANS:
PTS:
1
DIF:
Grade 12
REF: Lesson 4.2
OBJ: 5.1 Represent the number of arrangements of n elements taken n at a time, using factorial
notation. | 5.2 Determine, with or without technology, the value of a factorial. | 5.3 Simplify a
numeric or algebraic fraction containing factorials in both the numerator and denominator. | 5.4
Solve an equation that involves factorials.
TOP: Introducing Permutations and Factorial Notation
KEY: permutation | factorial notation
25. ANS:
6! = 720
PTS: 1
DIF: Grade 12
REF: Lesson 4.2
OBJ: 5.1 Represent the number of arrangements of n elements taken n at a time, using factorial
notation. | 5.2 Determine, with or without technology, the value of a factorial. | 5.3 Simplify a
numeric or algebraic fraction containing factorials in both the numerator and denominator. | 5.4
Solve an equation that involves factorials.
TOP: Introducing Permutations and Factorial Notation
KEY: permutation | factorial notation
26. ANS:
7! = 5040
PTS: 1
DIF: Grade 12
REF: Lesson 4.2
OBJ: 5.1 Represent the number of arrangements of n elements taken n at a time, using factorial
notation. | 5.2 Determine, with or without technology, the value of a factorial. | 5.3 Simplify a
numeric or algebraic fraction containing factorials in both the numerator and denominator. | 5.4
Solve an equation that involves factorials.
TOP: Introducing Permutations and Factorial Notation
KEY: permutation | factorial notation
27. ANS:
10
PTS: 1
DIF: Grade 12
REF: Lesson 4.2
OBJ: 5.1 Represent the number of arrangements of n elements taken n at a time, using factorial
notation. | 5.2 Determine, with or without technology, the value of a factorial. | 5.3 Simplify a
numeric or algebraic fraction containing factorials in both the numerator and denominator. | 5.4
Solve an equation that involves factorials.
TOP: Introducing Permutations and Factorial Notation
KEY: permutation | factorial notation
28. ANS:
13
PTS: 1
DIF: Grade 12
REF: Lesson 4.2
OBJ: 5.1 Represent the number of arrangements of n elements taken n at a time, using factorial
notation. | 5.2 Determine, with or without technology, the value of a factorial. | 5.3 Simplify a
numeric or algebraic fraction containing factorials in both the numerator and denominator. | 5.4
Solve an equation that involves factorials.
TOP: Introducing Permutations and Factorial Notation
KEY: permutation | factorial notation
29. ANS:
4
PTS: 1
DIF: Grade 12
REF: Lesson 4.2
OBJ: 5.1 Represent the number of arrangements of n elements taken n at a time, using factorial
notation. | 5.2 Determine, with or without technology, the value of a factorial. | 5.3 Simplify a
numeric or algebraic fraction containing factorials in both the numerator and denominator. | 5.4
Solve an equation that involves factorials.
TOP: Introducing Permutations and Factorial Notation
KEY: permutation | factorial notation
30. ANS:
3
PTS: 1
DIF: Grade 12
REF: Lesson 4.2
OBJ: 5.1 Represent the number of arrangements of n elements taken n at a time, using factorial
notation. | 5.2 Determine, with or without technology, the value of a factorial. | 5.3 Simplify a
numeric or algebraic fraction containing factorials in both the numerator and denominator. | 5.4
Solve an equation that involves factorials.
TOP: Introducing Permutations and Factorial Notation
KEY: permutation | factorial notation
31. ANS:
720
PTS: 1
DIF: Grade 12
REF: Lesson 4.3
OBJ: 5.5 Determine the number of permutations of n elements taken r at a time. | 5.8 Generalize
strategies for determining the number of permutations of n elements taken r at a time.
TOP: Permutations When All Objects Are Distinguishable
KEY: permutation
32. ANS:
60
PTS: 1
DIF: Grade 12
REF: Lesson 4.3
OBJ: 5.5 Determine the number of permutations of n elements taken r at a time. | 5.8 Generalize
strategies for determining the number of permutations of n elements taken r at a time.
TOP: Permutations When All Objects Are Distinguishable
KEY: permutation
33. ANS:
100
PTS: 1
DIF: Grade 12
REF: Lesson 4.3
OBJ: 5.5 Determine the number of permutations of n elements taken r at a time. | 5.8 Generalize
strategies for determining the number of permutations of n elements taken r at a time.
TOP: Permutations When All Objects Are Distinguishable
KEY: permutation
34. ANS:
95 040
PTS: 1
DIF: Grade 12
REF: Lesson 4.3
OBJ: 5.5 Determine the number of permutations of n elements taken r at a time. | 5.8 Generalize
strategies for determining the number of permutations of n elements taken r at a time.
TOP: Permutations When All Objects Are Distinguishable
KEY: permutation
35. ANS:
100P70
PTS: 1
DIF: Grade 12
REF: Lesson 4.3
OBJ: 5.5 Determine the number of permutations of n elements taken r at a time. | 5.8 Generalize
strategies for determining the number of permutations of n elements taken r at a time.
TOP: Permutations When All Objects Are Distinguishable
KEY: permutation
36. ANS:
12P10
PTS: 1
DIF: Grade 12
REF: Lesson 4.3
OBJ: 5.5 Determine the number of permutations of n elements taken r at a time. | 5.8 Generalize
strategies for determining the number of permutations of n elements taken r at a time.
TOP: Permutations When All Objects Are Distinguishable
KEY: permutation
37. ANS:
504
PTS: 1
DIF: Grade 12
REF: Lesson 4.3
OBJ: 5.5 Determine the number of permutations of n elements taken r at a time. | 5.8 Generalize
strategies for determining the number of permutations of n elements taken r at a time.
TOP: Permutations When All Objects Are Distinguishable
KEY: permutation
38. ANS:
132
PTS: 1
DIF: Grade 12
REF: Lesson 4.3
OBJ: 5.5 Determine the number of permutations of n elements taken r at a time. | 5.8 Generalize
strategies for determining the number of permutations of n elements taken r at a time.
TOP: Permutations When All Objects Are Distinguishable
KEY: permutation
39. ANS:
60 480
PTS: 1
DIF: Grade 12
REF: Lesson 4.3
OBJ: 5.5 Determine the number of permutations of n elements taken r at a time. | 5.8 Generalize
strategies for determining the number of permutations of n elements taken r at a time.
TOP: Permutations When All Objects Are Distinguishable
KEY: permutation
40. ANS:
n = 10
PTS: 1
DIF: Grade 12
REF: Lesson 4.3
OBJ: 5.5 Determine the number of permutations of n elements taken r at a time. | 5.8 Generalize
strategies for determining the number of permutations of n elements taken r at a time.
TOP: Permutations When All Objects Are Distinguishable
KEY: permutation
41. ANS:
n = 12
PTS: 1
DIF: Grade 12
REF: Lesson 4.3
OBJ: 5.5 Determine the number of permutations of n elements taken r at a time. | 5.8 Generalize
strategies for determining the number of permutations of n elements taken r at a time.
TOP: Permutations When All Objects Are Distinguishable
KEY: permutation
42. ANS:
r=1
PTS: 1
DIF: Grade 12
REF: Lesson 4.3
OBJ: 5.5 Determine the number of permutations of n elements taken r at a time. | 5.8 Generalize
strategies for determining the number of permutations of n elements taken r at a time.
TOP: Permutations When All Objects Are Distinguishable
KEY: permutation
43. ANS:
r=4
PTS: 1
DIF: Grade 12
REF: Lesson 4.3
OBJ: 5.5 Determine the number of permutations of n elements taken r at a time. | 5.8 Generalize
strategies for determining the number of permutations of n elements taken r at a time.
TOP: Permutations When All Objects Are Distinguishable
KEY: permutation
44. ANS:
r=3
PTS: 1
DIF: Grade 12
REF: Lesson 4.3
OBJ: 5.5 Determine the number of permutations of n elements taken r at a time. | 5.8 Generalize
strategies for determining the number of permutations of n elements taken r at a time.
TOP: Permutations When All Objects Are Distinguishable
KEY: permutation
45. ANS:
nP0 = 1, nP1 = n, and nPn = n!
PTS: 1
DIF: Grade 12
REF: Lesson 4.3
OBJ: 5.5 Determine the number of permutations of n elements taken r at a time. | 5.8 Generalize
strategies for determining the number of permutations of n elements taken r at a time.
TOP: Permutations When All Objects Are Distinguishable
KEY: permutation | factorial notation
46. ANS:
10
PTS: 1
DIF: Grade 12
REF: Lesson 4.4
OBJ: 5.6 Determine the number of permutations of n elements taken n at a time where some
elements are not distinct. | 5.7 Explain, using examples, the effect on the total number of
permutations of n elements when two or more elements are identical.
TOP: Permutations When Objects Are Identical
KEY: permutation | factorial notation
47. ANS:
1120
PTS: 1
DIF: Grade 12
REF: Lesson 4.4
OBJ: 5.6 Determine the number of permutations of n elements taken n at a time where some
elements are not distinct. | 5.7 Explain, using examples, the effect on the total number of
permutations of n elements when two or more elements are identical.
TOP: Permutations When Objects Are Identical
KEY: permutation | factorial notation
48. ANS:
75 600
PTS: 1
DIF: Grade 12
REF: Lesson 4.4
OBJ: 5.6 Determine the number of permutations of n elements taken n at a time where some
elements are not distinct. | 5.7 Explain, using examples, the effect on the total number of
permutations of n elements when two or more elements are identical.
TOP: Permutations When Objects Are Identical
KEY: permutation | factorial notation
49. ANS:
120 120
PTS: 1
DIF: Grade 12
REF: Lesson 4.4
OBJ: 5.6 Determine the number of permutations of n elements taken n at a time where some
elements are not distinct. | 5.7 Explain, using examples, the effect on the total number of
permutations of n elements when two or more elements are identical.
TOP: Permutations When Objects Are Identical
KEY: permutation | factorial notation
50. ANS:
10 080
PTS: 1
DIF: Grade 12
REF: Lesson 4.4
OBJ: 5.6 Determine the number of permutations of n elements taken n at a time where some
elements are not distinct. | 5.7 Explain, using examples, the effect on the total number of
permutations of n elements when two or more elements are identical.
TOP: Permutations When Objects Are Identical
KEY: permutation | factorial notation
51. ANS:
181 440
PTS: 1
DIF: Grade 12
REF: Lesson 4.4
OBJ: 5.6 Determine the number of permutations of n elements taken n at a time where some
elements are not distinct. | 5.7 Explain, using examples, the effect on the total number of
permutations of n elements when two or more elements are identical.
TOP: Permutations When Objects Are Identical
KEY: permutation | factorial notation
52. ANS:
9 979 200
PTS:
1
DIF:
Grade 12
REF: Lesson 4.4
OBJ: 5.6 Determine the number of permutations of n elements taken n at a time where some
elements are not distinct. | 5.7 Explain, using examples, the effect on the total number of
permutations of n elements when two or more elements are identical.
TOP: Permutations When Objects Are Identical
KEY: permutation | factorial notation
53. ANS:
181 440
PTS: 1
DIF: Grade 12
REF: Lesson 4.4
OBJ: 5.6 Determine the number of permutations of n elements taken n at a time where some
elements are not distinct. | 5.7 Explain, using examples, the effect on the total number of
permutations of n elements when two or more elements are identical.
TOP: Permutations When Objects Are Identical
KEY: permutation | factorial notation
54. ANS:
56
PTS: 1
DIF: Grade 12
REF: Lesson 4.4
OBJ: 5.6 Determine the number of permutations of n elements taken n at a time where some
elements are not distinct. | 5.7 Explain, using examples, the effect on the total number of
permutations of n elements when two or more elements are identical.
TOP: Permutations When Objects Are Identical
KEY: permutation | factorial notation
55. ANS:
100
PTS: 1
DIF: Grade 12
REF: Lesson 4.4
OBJ: 5.6 Determine the number of permutations of n elements taken n at a time where some
elements are not distinct. | 5.7 Explain, using examples, the effect on the total number of
permutations of n elements when two or more elements are identical.
TOP: Permutations When Objects Are Identical
KEY: permutation | factorial notation
56. ANS:
120
PTS: 1
DIF: Grade 12
REF: Lesson 4.4
OBJ: 5.6 Determine the number of permutations of n elements taken n at a time where some
elements are not distinct. | 5.7 Explain, using examples, the effect on the total number of
permutations of n elements when two or more elements are identical.
TOP: Permutations When Objects Are Identical
KEY: permutation | factorial notation
57. ANS:
525
PTS: 1
DIF: Grade 12
REF: Lesson 4.4
OBJ: 5.6 Determine the number of permutations of n elements taken n at a time where some
elements are not distinct. | 5.7 Explain, using examples, the effect on the total number of
permutations of n elements when two or more elements are identical.
TOP: Permutations When Objects Are Identical
KEY: permutation | factorial notation
58. ANS:
924
PTS: 1
DIF: Grade 12
REF: Lesson 4.4
OBJ: 5.6 Determine the number of permutations of n elements taken n at a time where some
elements are not distinct. | 5.7 Explain, using examples, the effect on the total number of
permutations of n elements when two or more elements are identical.
TOP: Permutations When Objects Are Identical
KEY: permutation | factorial notation
59. ANS:
210
PTS: 1
DIF: Grade 12
REF: Lesson 4.4
OBJ: 5.6 Determine the number of permutations of n elements taken n at a time where some
elements are not distinct. | 5.7 Explain, using examples, the effect on the total number of
permutations of n elements when two or more elements are identical.
TOP: Permutations When Objects Are Identical
KEY: permutation | factorial notation
60. ANS:
176
PTS: 1
DIF: Grade 12
REF: Lesson 4.4
OBJ: 5.6 Determine the number of permutations of n elements taken n at a time where some
elements are not distinct. | 5.7 Explain, using examples, the effect on the total number of
permutations of n elements when two or more elements are identical.
TOP: Permutations When Objects Are Identical
KEY: permutation | factorial notation
61. ANS:
792
PTS: 1
DIF: Grade 12
REF: Lesson 4.5
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 5.1 Represent
the number of arrangements of n elements taken n at a time, using factorial notation.
TOP: Exploring Combinations
KEY: counting | combination | factorial notation
62. ANS:
120
PTS: 1
DIF: Grade 12
REF: Lesson 4.5
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 5.1 Represent
the number of arrangements of n elements taken n at a time, using factorial notation.
TOP: Exploring Combinations
KEY: counting | combination | factorial notation
63. ANS:
36
PTS: 1
DIF: Grade 12
REF: Lesson 4.5
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 5.1 Represent
the number of arrangements of n elements taken n at a time, using factorial notation.
TOP: Exploring Combinations
KEY: counting | combination | factorial notation
64. ANS:
28
PTS: 1
DIF: Grade 12
REF: Lesson 4.5
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 5.1 Represent
the number of arrangements of n elements taken n at a time, using factorial notation.
TOP: Exploring Combinations
KEY: counting | combination | factorial notation
65. ANS:
84
PTS: 1
DIF: Grade 12
REF: Lesson 4.5
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 5.1 Represent
the number of arrangements of n elements taken n at a time, using factorial notation.
TOP: Exploring Combinations
KEY: counting | combination | factorial notation
66. ANS:
10
PTS: 1
DIF: Grade 12
REF: Lesson 4.6
OBJ: 6.1 Explain, using examples, why order is or is not important when solving problems that
involve permutations or combinations. | 6.2 Determine the number of combinations of n elements
taken r at a time. | 6.3 Generalize strategies for determining the number of combinations of n
elements taken r at a time.
TOP: Combinations
KEY: counting | combination
67. ANS:
495
PTS: 1
DIF: Grade 12
REF: Lesson 4.6
OBJ: 6.1 Explain, using examples, why order is or is not important when solving problems that
involve permutations or combinations. | 6.2 Determine the number of combinations of n elements
taken r at a time. | 6.3 Generalize strategies for determining the number of combinations of n
elements taken r at a time.
TOP: Combinations
KEY: counting | combination
68. ANS:
5
PTS: 1
DIF: Grade 12
REF: Lesson 4.6
OBJ: 6.1 Explain, using examples, why order is or is not important when solving problems that
involve permutations or combinations. | 6.2 Determine the number of combinations of n elements
taken r at a time. | 6.3 Generalize strategies for determining the number of combinations of n
elements taken r at a time.
TOP: Combinations
KEY: counting | combination
69. ANS:
21
PTS: 1
DIF: Grade 12
REF: Lesson 4.6
OBJ: 6.1 Explain, using examples, why order is or is not important when solving problems that
involve permutations or combinations. | 6.2 Determine the number of combinations of n elements
taken r at a time. | 6.3 Generalize strategies for determining the number of combinations of n
elements taken r at a time.
TOP: Combinations
KEY: counting | combination
70. ANS:
120
PTS: 1
DIF: Grade 12
REF: Lesson 4.6
OBJ: 6.1 Explain, using examples, why order is or is not important when solving problems that
involve permutations or combinations. | 6.2 Determine the number of combinations of n elements
taken r at a time. | 6.3 Generalize strategies for determining the number of combinations of n
elements taken r at a time.
TOP: Combinations
KEY: counting | combination
71. ANS:
4495
PTS: 1
DIF: Grade 12
REF: Lesson 4.6
OBJ: 6.1 Explain, using examples, why order is or is not important when solving problems that
involve permutations or combinations. | 6.2 Determine the number of combinations of n elements
taken r at a time. | 6.3 Generalize strategies for determining the number of combinations of n
elements taken r at a time.
TOP: Combinations
KEY: counting | combination
72. ANS:
13 983 816
PTS: 1
DIF: Grade 12
REF: Lesson 4.6
OBJ: 6.1 Explain, using examples, why order is or is not important when solving problems that
involve permutations or combinations. | 6.2 Determine the number of combinations of n elements
taken r at a time. | 6.3 Generalize strategies for determining the number of combinations of n
elements taken r at a time.
TOP: Combinations
KEY: counting | combination
73. ANS:
660
PTS: 1
DIF: Grade 12
REF: Lesson 4.6
OBJ: 6.1 Explain, using examples, why order is or is not important when solving problems that
involve permutations or combinations. | 6.2 Determine the number of combinations of n elements
taken r at a time. | 6.3 Generalize strategies for determining the number of combinations of n
elements taken r at a time.
TOP: Combinations
KEY: counting | combination | Fundamental Counting Principle
74. ANS:
2730
PTS: 1
DIF: Grade 12
REF: Lesson 4.6
OBJ: 6.1 Explain, using examples, why order is or is not important when solving problems that
involve permutations or combinations. | 6.2 Determine the number of combinations of n elements
taken r at a time. | 6.3 Generalize strategies for determining the number of combinations of n
elements taken r at a time.
TOP: Combinations
KEY: counting | combination | Fundamental Counting Principle
75. ANS:
280
PTS: 1
DIF: Grade 12
REF: Lesson 4.6
OBJ: 6.1 Explain, using examples, why order is or is not important when solving problems that
involve permutations or combinations. | 6.2 Determine the number of combinations of n elements
taken r at a time. | 6.3 Generalize strategies for determining the number of combinations of n
elements taken r at a time.
TOP: Combinations
KEY: counting | combination | Fundamental Counting Principle
76. ANS:
360
PTS: 1
DIF: Grade 12
REF: Lesson 4.6
OBJ: 6.1 Explain, using examples, why order is or is not important when solving problems that
involve permutations or combinations. | 6.2 Determine the number of combinations of n elements
taken r at a time. | 6.3 Generalize strategies for determining the number of combinations of n
elements taken r at a time.
TOP: Combinations
KEY: counting | combination | Fundamental Counting Principle
77. ANS:
n = 10
PTS: 1
DIF: Grade 12
REF: Lesson 4.6
OBJ: 6.1 Explain, using examples, why order is or is not important when solving problems that
involve permutations or combinations. | 6.2 Determine the number of combinations of n elements
taken r at a time. | 6.3 Generalize strategies for determining the number of combinations of n
elements taken r at a time.
TOP: Combinations
KEY: counting | combination
78. ANS:
n=8
PTS: 1
DIF: Grade 12
REF: Lesson 4.6
OBJ: 6.1 Explain, using examples, why order is or is not important when solving problems that
involve permutations or combinations. | 6.2 Determine the number of combinations of n elements
taken r at a time. | 6.3 Generalize strategies for determining the number of combinations of n
elements taken r at a time.
TOP: Combinations
KEY: counting | combination
79. ANS:
r = 34 or r = 0
PTS: 1
DIF: Grade 12
REF: Lesson 4.6
OBJ: 6.1 Explain, using examples, why order is or is not important when solving problems that
involve permutations or combinations. | 6.2 Determine the number of combinations of n elements
taken r at a time. | 6.3 Generalize strategies for determining the number of combinations of n
elements taken r at a time.
TOP: Combinations
KEY: counting | combination
80. ANS:
r=3
PTS: 1
DIF: Grade 12
REF: Lesson 4.6
OBJ: 6.1 Explain, using examples, why order is or is not important when solving problems that
involve permutations or combinations. | 6.2 Determine the number of combinations of n elements
taken r at a time. | 6.3 Generalize strategies for determining the number of combinations of n
elements taken r at a time.
TOP: Combinations
KEY: counting | combination
81. ANS:
10 080
PTS: 1
DIF: Grade 12
REF: Lesson 4.7
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 4.4 Solve a
contextual counting problem, using the fundamental counting principle, and explain the reasoning.
| 5.5 Determine the number of permutations of n elements taken r at a time. | 6.2 Determine the
number of combinations of n elements taken r at a time.
TOP: Solving Counting Problems
KEY: counting | permutation
82. ANS:
1440
PTS: 1
DIF: Grade 12
REF: Lesson 4.7
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 4.4 Solve a
contextual counting problem, using the fundamental counting principle, and explain the reasoning.
| 5.5 Determine the number of permutations of n elements taken r at a time. | 6.2 Determine the
number of combinations of n elements taken r at a time.
TOP: Solving Counting Problems
KEY: counting | permutation
83. ANS:
9 465 511 770
PTS: 1
DIF: Grade 12
REF: Lesson 4.7
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 4.4 Solve a
contextual counting problem, using the fundamental counting principle, and explain the reasoning.
| 5.5 Determine the number of permutations of n elements taken r at a time. | 6.2 Determine the
number of combinations of n elements taken r at a time.
TOP: Solving Counting Problems
KEY: counting | combination | Fundamental Counting Principle
84. ANS:
2 308 743 493 056
PTS: 1
DIF: Grade 12
REF: Lesson 4.7
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 4.4 Solve a
contextual counting problem, using the fundamental counting principle, and explain the reasoning.
| 5.5 Determine the number of permutations of n elements taken r at a time. | 6.2 Determine the
number of combinations of n elements taken r at a time.
TOP: Solving Counting Problems
KEY: counting | combination | Fundamental Counting Principle
85. ANS:
96 197 645 544
PTS: 1
DIF: Grade 12
REF: Lesson 4.7
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 4.4 Solve a
contextual counting problem, using the fundamental counting principle, and explain the reasoning.
| 5.5 Determine the number of permutations of n elements taken r at a time. | 6.2 Determine the
number of combinations of n elements taken r at a time.
TOP: Solving Counting Problems
KEY: counting | combination | Fundamental Counting Principle
86. ANS:
66 512 160
PTS: 1
DIF: Grade 12
REF: Lesson 4.7
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 4.4 Solve a
contextual counting problem, using the fundamental counting principle, and explain the reasoning.
| 5.5 Determine the number of permutations of n elements taken r at a time. | 6.2 Determine the
number of combinations of n elements taken r at a time.
TOP: Solving Counting Problems
KEY: counting | combination | Fundamental Counting Principle
87. ANS:
4845
PTS: 1
DIF: Grade 12
REF: Lesson 4.7
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 4.4 Solve a
contextual counting problem, using the fundamental counting principle, and explain the reasoning.
| 5.5 Determine the number of permutations of n elements taken r at a time. | 6.2 Determine the
number of combinations of n elements taken r at a time.
TOP: Solving Counting Problems
KEY: counting | combination | Fundamental Counting Principle
88. ANS:
1221
PTS: 1
DIF: Grade 12
REF: Lesson 4.7
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 4.4 Solve a
contextual counting problem, using the fundamental counting principle, and explain the reasoning.
| 5.5 Determine the number of permutations of n elements taken r at a time. | 6.2 Determine the
number of combinations of n elements taken r at a time.
TOP: Solving Counting Problems
KEY: counting | combination | Fundamental Counting Principle
89. ANS:
6732
PTS: 1
DIF: Grade 12
REF: Lesson 4.7
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 4.4 Solve a
contextual counting problem, using the fundamental counting principle, and explain the reasoning.
| 5.5 Determine the number of permutations of n elements taken r at a time. | 6.2 Determine the
number of combinations of n elements taken r at a time.
TOP: Solving Counting Problems
KEY: counting | combination | Fundamental Counting Principle
90. ANS:
21 808
PTS: 1
DIF: Grade 12
REF: Lesson 4.7
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 4.4 Solve a
contextual counting problem, using the fundamental counting principle, and explain the reasoning.
| 5.5 Determine the number of permutations of n elements taken r at a time. | 6.2 Determine the
number of combinations of n elements taken r at a time.
TOP: Solving Counting Problems
KEY: counting | combination | Fundamental Counting Principle
91. ANS:
258 856
PTS: 1
DIF: Grade 12
REF: Lesson 4.7
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 4.4 Solve a
contextual counting problem, using the fundamental counting principle, and explain the reasoning.
| 5.5 Determine the number of permutations of n elements taken r at a time. | 6.2 Determine the
number of combinations of n elements taken r at a time.
TOP: Solving Counting Problems
KEY: counting | combination | Fundamental Counting Principle
92. ANS:
45
PTS: 1
DIF: Grade 12
REF: Lesson 4.7
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 4.4 Solve a
contextual counting problem, using the fundamental counting principle, and explain the reasoning.
| 5.5 Determine the number of permutations of n elements taken r at a time. | 6.2 Determine the
number of combinations of n elements taken r at a time.
TOP: Solving Counting Problems
KEY: counting | combination
93. ANS:
462
PTS:
1
DIF:
Grade 12
REF: Lesson 4.7
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 4.4 Solve a
contextual counting problem, using the fundamental counting principle, and explain the reasoning.
| 5.5 Determine the number of permutations of n elements taken r at a time. | 6.2 Determine the
number of combinations of n elements taken r at a time.
TOP: Solving Counting Problems
KEY: counting | combination
94. ANS:
506 250 000
PTS: 1
DIF: Grade 12
REF: Lesson 4.7
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 4.4 Solve a
contextual counting problem, using the fundamental counting principle, and explain the reasoning.
| 5.5 Determine the number of permutations of n elements taken r at a time. | 6.2 Determine the
number of combinations of n elements taken r at a time.
TOP: Solving Counting Problems
KEY: counting | Fundamental Counting Principle
95. ANS:
486 246 600
PTS: 1
DIF: Grade 12
REF: Lesson 4.7
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 4.4 Solve a
contextual counting problem, using the fundamental counting principle, and explain the reasoning.
| 5.5 Determine the number of permutations of n elements taken r at a time. | 6.2 Determine the
number of combinations of n elements taken r at a time.
TOP: Solving Counting Problems
KEY: counting | permutation
PROBLEM
1. ANS:
a)
There are 12 different outcomes.
b) The number of outcomes, O, is related to the number of sides on the die and the number of sides
on a coin:
O = (number of sides on die)  (number of sides on coin)
O=62
O = 12
There are 12 different outcomes.
PTS: 1
DIF: Grade 12
REF: Lesson 4.1
OBJ: 4.1 Represent and solve counting problems, using a graphic organizer. | 4.2 Generalize the
fundamental counting principle, using inductive reasoning. | 4.3 Identify and explain assumptions
made in solving a counting problem. | 4.4 Solve a contextual counting problem, using the
fundamental counting principle, and explain the reasoning.
TOP: Counting Principles
KEY: counting | Fundamental Counting Principle
2. ANS:
a)
There are 8 different variations of the hockey uniform to choose from.
b) The number of uniform variations, U, is related to the number of sweaters and the number of
pants:
U = (number of sweaters)  (number of pants)
U=42
U=8
There are 8 different variations of the hockey uniform to choose from.
PTS: 1
DIF: Grade 12
REF: Lesson 4.1
OBJ: 4.1 Represent and solve counting problems, using a graphic organizer. | 4.2 Generalize the
fundamental counting principle, using inductive reasoning. | 4.3 Identify and explain assumptions
made in solving a counting problem. | 4.4 Solve a contextual counting problem, using the
fundamental counting principle, and explain the reasoning.
TOP: Counting Principles
KEY: counting | Fundamental Counting Principle
3. ANS:
a)
There are 15 different colour–size variations.
b) The number of colour–size variations, C, is related to the number of colours and the number of
sizes:
C = (number of colours)  (number of sizes)
C=35
C = 15
There are 15 different colour–size variations.
PTS: 1
DIF: Grade 12
REF: Lesson 4.1
OBJ: 4.1 Represent and solve counting problems, using a graphic organizer. | 4.2 Generalize the
fundamental counting principle, using inductive reasoning. | 4.3 Identify and explain assumptions
made in solving a counting problem. | 4.4 Solve a contextual counting problem, using the
fundamental counting principle, and explain the reasoning.
TOP: Counting Principles
KEY: counting | Fundamental Counting Principle
4. ANS:
a) The number of different codes, C, is related to the number of digits from which to select on each
wheel of the lock, D:
C = D1 D2  D3  D4
C=8888
C = 4096
There are 4096 different four-digit codes on this type of lock.
b) The number of different codes, N, is related to the number of numbers from which to select on
each wheel of the lock, W:
N = W1  W2  W3  W4
N=8765
N = 1680
There are only 1680 different four-digit codes when the digits cannot repeat.
PTS: 1
DIF: Grade 12
REF: Lesson 4.1
OBJ: 4.1 Represent and solve counting problems, using a graphic organizer. | 4.2 Generalize the
fundamental counting principle, using inductive reasoning. | 4.3 Identify and explain assumptions
made in solving a counting problem. | 4.4 Solve a contextual counting problem, using the
fundamental counting principle, and explain the reasoning.
TOP: Counting Principles
KEY: counting | Fundamental Counting Principle
5. ANS:
a) The number of different codes, C, is related to the number of digits from which to select on each
wheel of the lock, D:
C = D1  D2  D3  D4  D5  D6
C = 10  10  10  10  10  10
C = 1 000 000
There are 1 000 000 different six-digit codes on this type of lock.
b) First determine the number of codes without repetition.
The number of different codes, N, is related to the number of digits from which to select on each
wheel of the lock, W:
N = W1  W2  W3  W4  W5  W6
N = 10  9  8  7  6  5
N = 151 200
Approximately 15% of these codes have no repeated digits.
PTS: 1
DIF: Grade 12
REF: Lesson 4.1
OBJ: 4.1 Represent and solve counting problems, using a graphic organizer. | 4.2 Generalize the
fundamental counting principle, using inductive reasoning. | 4.3 Identify and explain assumptions
made in solving a counting problem. | 4.4 Solve a contextual counting problem, using the
fundamental counting principle, and explain the reasoning.
TOP: Counting Principles
KEY: counting | Fundamental Counting Principle
6. ANS:
a) There are 20 letters from A to T.
The number of different codes, C, is related to the number of letters from which to select on each
wheel of the lock, W:
C = W1  W2  W3
C = 20  19  18
C = 6840
There are 6840 different three-letter codes on this type of lock.
b) The number of different codes, R, is related to the number of letters from which to select on
each wheel of the lock, X:
R = X1  X2  X3
R = 20  20  20
R = 8000
R – C = 8000 – 6840
R – C = 1160
There are 1160 more codes if repetition is allowed.
PTS:
1
DIF:
Grade 12
REF: Lesson 4.1
OBJ: 4.1 Represent and solve counting problems, using a graphic organizer. | 4.2 Generalize the
fundamental counting principle, using inductive reasoning. | 4.3 Identify and explain assumptions
made in solving a counting problem. | 4.4 Solve a contextual counting problem, using the
fundamental counting principle, and explain the reasoning.
TOP: Counting Principles
KEY: counting | Fundamental Counting Principle
7. ANS:
a) There are 12 letters from A to L.
The number of different codes, C, is related to the number of letters from which to select on each
wheel of the lock, W:
C = W1  W2  W3  W4
C = 12  12  12  12
C = 20 736
There are 20 736 different four-letter codes on this type of lock.
b) First determine the number of codes without repetition.
The number of different codes, N, is related to the number of letters from which to select on each
wheel of the lock, X:
N = X1  X2  X3  X4
N = 12  11  10  9
N = 11 880
C – N = 20 736 – 11 880
C – N = 8856
Approximately 43% of these codes repeat at least one letter.
PTS: 1
DIF: Grade 12
REF: Lesson 4.1
OBJ: 4.1 Represent and solve counting problems, using a graphic organizer. | 4.2 Generalize the
fundamental counting principle, using inductive reasoning. | 4.3 Identify and explain assumptions
made in solving a counting problem. | 4.4 Solve a contextual counting problem, using the
fundamental counting principle, and explain the reasoning.
TOP: Counting Principles
KEY: counting | Fundamental Counting Principle
8. ANS:
a) Event A: Draw a jack.
OR
Event B: Draw a red ace.
There are 4 jacks in a euchre deck: n(A) = 4
There are 2 red aces in a euchre deck: n(B) = 2
These events are mutually exclusive.
n(A  B) = n(A) + n(B)
n(A  B) = 4 + 2
n(A  B) = 6
There are 6 ways to draw a single card and get either a jack or a red ace.
b) Event C: Draw a number card.
OR
Event D: Draw a heart.
There are 8 number cards in a euchre deck: n(C) = 8
There are 6 hearts in a euchre deck: n(D) = 6
There are 2 number cards that are also hearts in a euchre deck: n(C  D) = 2
n(C  D) = n(C) + n(D) – n(C  D)
n(C  D) = 8 + 6 – 2
n(C  D) = 12
There are 12 ways to draw a single card and get either a number card or a heart.
PTS: 1
DIF: Grade 12
REF: Lesson 4.1
OBJ: 4.1 Represent and solve counting problems, using a graphic organizer. | 4.2 Generalize the
fundamental counting principle, using inductive reasoning. | 4.3 Identify and explain assumptions
made in solving a counting problem. | 4.4 Solve a contextual counting problem, using the
fundamental counting principle, and explain the reasoning.
TOP: Counting Principles
KEY: counting | Fundamental Counting Principle
9. ANS:
a) Event A: Draw a face card.
OR
Event B: Draw a red card.
There are 12 face cards in a euchre deck: n(A) = 12
There are 12 red cards in a euchre deck: n(B) = 12
There are 6 red face cards in a euchre deck: n(A  B) = 6
n(A  B) = n(A) + n(B) – n(A  B)
n(A  B) = 12 + 12 – 6
n(A  B) = 18
There are 18 ways to draw a single card and get either a face card or a red card.
b) No, the Fundamental Counting Principle does not apply to this situation. There is only one task
being performed.
PTS: 1
DIF: Grade 12
REF: Lesson 4.1
OBJ: 4.1 Represent and solve counting problems, using a graphic organizer. | 4.2 Generalize the
fundamental counting principle, using inductive reasoning. | 4.3 Identify and explain assumptions
made in solving a counting problem. | 4.4 Solve a contextual counting problem, using the
fundamental counting principle, and explain the reasoning.
TOP: Counting Principles
KEY: counting | Fundamental Counting Principle
10. ANS:
Event A: Draw a face card.
OR
Event B: Draw a spade.
There are 12 face cards: n(A) = 12
There are 13 red cards: n(B) = 13
There are 3 face cards that are spades: n(A  B) = 3
n(A  B) = n(A) + n(B) – n(A  B)
n(A  B) = 12 + 13 – 3
n(A  B) = 22
There are 22 ways to draw a single card and get either a face card or a spade.
There are 4 ways to roll a four-sided die, D.
The number of different outcomes, O, is related to A  B and D:
O = n(A  B)  n(D)
O = 22  4
O = 88
There are 88 different ways to roll the die and draw either a face card or a spade.
PTS: 1
DIF: Grade 12
REF: Lesson 4.1
OBJ: 4.1 Represent and solve counting problems, using a graphic organizer. | 4.2 Generalize the
fundamental counting principle, using inductive reasoning. | 4.3 Identify and explain assumptions
made in solving a counting problem. | 4.4 Solve a contextual counting problem, using the
fundamental counting principle, and explain the reasoning.
TOP: Counting Principles
KEY: counting | Fundamental Counting Principle
11. ANS:
PTS: 1
DIF: Grade 12
REF: Lesson 4.2
OBJ: 5.1 Represent the number of arrangements of n elements taken n at a time, using factorial
notation. | 5.2 Determine, with or without technology, the value of a factorial. | 5.3 Simplify a
numeric or algebraic fraction containing factorials in both the numerator and denominator. | 5.4
Solve an equation that involves factorials.
TOP: Introducing Permutations and Factorial Notation
KEY: counting | permutation | factorial notation
12. ANS:
PTS: 1
DIF: Grade 12
REF: Lesson 4.2
OBJ: 5.1 Represent the number of arrangements of n elements taken n at a time, using factorial
notation. | 5.2 Determine, with or without technology, the value of a factorial. | 5.3 Simplify a
numeric or algebraic fraction containing factorials in both the numerator and denominator. | 5.4
Solve an equation that involves factorials.
TOP: Introducing Permutations and Factorial Notation
KEY: counting | permutation | factorial notation
13. ANS:
No. In the expression n!, the variable n is defined only for values that belong to the set of natural
numbers.
PTS: 1
DIF: Grade 12
REF: Lesson 4.2
OBJ: 5.1 Represent the number of arrangements of n elements taken n at a time, using factorial
notation. | 5.2 Determine, with or without technology, the value of a factorial. | 5.3 Simplify a
numeric or algebraic fraction containing factorials in both the numerator and denominator. | 5.4
Solve an equation that involves factorials.
TOP: Introducing Permutations and Factorial Notation
KEY: counting | permutation | factorial notation
14. ANS:
No. In the expression n!, the variable n is defined only for values that belong to the set of natural
numbers.
PTS: 1
DIF: Grade 12
REF: Lesson 4.2
OBJ: 5.1 Represent the number of arrangements of n elements taken n at a time, using factorial
notation. | 5.2 Determine, with or without technology, the value of a factorial. | 5.3 Simplify a
numeric or algebraic fraction containing factorials in both the numerator and denominator. | 5.4
Solve an equation that involves factorials.
TOP: Introducing Permutations and Factorial Notation
KEY: counting | permutation | factorial notation
15. ANS:
a) Answers may vary. Sample answer: SBRUNME, NUMBRES, and BRUMENS
b) There are 7! possible permutations because there are 7 letters and 7 positions for them to
occupy.
PTS: 1
DIF: Grade 12
REF: Lesson 4.2
OBJ: 5.1 Represent the number of arrangements of n elements taken n at a time, using factorial
notation. | 5.2 Determine, with or without technology, the value of a factorial. | 5.3 Simplify a
numeric or algebraic fraction containing factorials in both the numerator and denominator. | 5.4
Solve an equation that involves factorials.
TOP: Introducing Permutations and Factorial Notation
KEY: counting | permutation | factorial notation
16. ANS:
a) Answers may vary. Sample answer: EPERMUT, EEMPRTU, and EMPRUTE
b) There are 6! possible permutations because there are 6 letters (if one of the E’s is always the
first letter) and 6 positions for them to occupy.
PTS: 1
DIF: Grade 12
REF: Lesson 4.2
OBJ: 5.1 Represent the number of arrangements of n elements taken n at a time, using factorial
notation. | 5.2 Determine, with or without technology, the value of a factorial. | 5.3 Simplify a
numeric or algebraic fraction containing factorials in both the numerator and denominator. | 5.4
Solve an equation that involves factorials.
TOP: Introducing Permutations and Factorial Notation
KEY: counting | permutation | factorial notation
17. ANS:
Neither value is greater than the other: 6! =
.
PTS: 1
DIF: Grade 12
REF: Lesson 4.2
OBJ: 5.1 Represent the number of arrangements of n elements taken n at a time, using factorial
notation. | 5.2 Determine, with or without technology, the value of a factorial. | 5.3 Simplify a
numeric or algebraic fraction containing factorials in both the numerator and denominator. | 5.4
Solve an equation that involves factorials.
TOP: Introducing Permutations and Factorial Notation
KEY: counting | permutation | factorial notation
18. ANS:
B is slightly greater than A:
<
.
PTS: 1
DIF: Grade 12
REF: Lesson 4.2
OBJ: 5.1 Represent the number of arrangements of n elements taken n at a time, using factorial
notation. | 5.2 Determine, with or without technology, the value of a factorial. | 5.3 Simplify a
numeric or algebraic fraction containing factorials in both the numerator and denominator. | 5.4
Solve an equation that involves factorials.
TOP: Introducing Permutations and Factorial Notation
KEY: counting | permutation | factorial notation
19. ANS:
A is slightly greater than B: 10! >
.
PTS: 1
DIF: Grade 12
REF: Lesson 4.2
OBJ: 5.1 Represent the number of arrangements of n elements taken n at a time, using factorial
notation. | 5.2 Determine, with or without technology, the value of a factorial. | 5.3 Simplify a
numeric or algebraic fraction containing factorials in both the numerator and denominator. | 5.4
Solve an equation that involves factorials.
TOP: Introducing Permutations and Factorial Notation
KEY: counting | permutation | factorial notation
20. ANS:
There are 3! or 6 possible orders for completing the morning activities.
There are 4! or 24 possible orders for completing the afternoon activities.
By the Fundamental Counting Principle, there are 6  24 or 144 ways to complete all 7 activities, if
you complete the morning activities first.
PTS: 1
DIF: Grade 12
REF: Lesson 4.2
OBJ: 5.1 Represent the number of arrangements of n elements taken n at a time, using factorial
notation. | 5.2 Determine, with or without technology, the value of a factorial. | 5.3 Simplify a
numeric or algebraic fraction containing factorials in both the numerator and denominator. | 5.4
Solve an equation that involves factorials.
TOP: Introducing Permutations and Factorial Notation
KEY: counting | permutation | factorial notation | Fundamental Counting Principle
21. ANS:
There are 8 cars and 6 positions they can be placed in.
Let A represent the number of arrangements:
The cars can be parked 20 160 different ways.
PTS: 1
DIF: Grade 12
REF: Lesson 4.3
OBJ: 5.5 Determine the number of permutations of n elements taken r at a time. | 5.8 Generalize
strategies for determining the number of permutations of n elements taken r at a time.
TOP: Permutations When All Objects Are Distinguishable
KEY: counting | permutation | factorial notation
22. ANS:
There are 11 cars and 5 positions they can be placed in.
Let A represent the number of arrangements:
The cars can be parked 55 440 different ways.
PTS: 1
DIF: Grade 12
REF: Lesson 4.3
OBJ: 5.5 Determine the number of permutations of n elements taken r at a time. | 5.8 Generalize
strategies for determining the number of permutations of n elements taken r at a time.
TOP: Permutations When All Objects Are Distinguishable
KEY: counting | permutation | factorial notation
23. ANS:
There are 4 red cars and 2 positions they can be placed in.
There are 6 silver cars and 3 positions they can be placed in.
Let A represent the number of arrangements:
The cars can be parked 1440 different ways.
PTS: 1
DIF: Grade 12
REF: Lesson 4.3
OBJ: 5.5 Determine the number of permutations of n elements taken r at a time. | 5.8 Generalize
strategies for determining the number of permutations of n elements taken r at a time.
TOP: Permutations When All Objects Are Distinguishable
KEY: counting | permutation | factorial notation
24. ANS:
There are 3 red cars and 2 positions they can be placed in.
There are 8 silver cars and 4 positions they can be placed in.
Let A represent the number of arrangements:
The cars can be parked 10 080 different ways.
PTS: 1
DIF: Grade 12
REF: Lesson 4.3
OBJ: 5.5 Determine the number of permutations of n elements taken r at a time. | 5.8 Generalize
strategies for determining the number of permutations of n elements taken r at a time.
TOP: Permutations When All Objects Are Distinguishable
KEY: counting | permutation | factorial notation
25. ANS:
There are 12 songs and 5 positions they can be placed in.
Let A represent the number of arrangements:
There are 95 040 different 5-song playlists that can be created from 12 songs
PTS: 1
DIF: Grade 12
REF: Lesson 4.3
OBJ: 5.5 Determine the number of permutations of n elements taken r at a time. | 5.8 Generalize
strategies for determining the number of permutations of n elements taken r at a time.
TOP: Permutations When All Objects Are Distinguishable
KEY: counting | permutation | factorial notation
26. ANS:
a) There are 140 songs and 3 positions they can be placed in.
Let A represent the number of arrangements:
There are 2 685 480 possible sets of three songs, without repetition.
b) With repetition, there are 140 possibilities for each position in the first three songs.
There are 2 744 000 possible sets of three songs, with repetition.
PTS: 1
DIF: Grade 12
REF: Lesson 4.3
OBJ: 5.5 Determine the number of permutations of n elements taken r at a time. | 5.8 Generalize
strategies for determining the number of permutations of n elements taken r at a time.
TOP: Permutations When All Objects Are Distinguishable
KEY: counting | permutation | factorial notation
27. ANS:
a) There are 172 songs and 4 positions they can be placed in.
Let A represent the number of arrangements:
There are 845 006 760 possible sets of four songs, without repetition.
b) With repetition, there are 172 possibilities for each position in the first four songs.
There are 875 213 056 possible sets of four songs, with repetition.
PTS: 1
DIF: Grade 12
REF: Lesson 4.3
OBJ: 5.5 Determine the number of permutations of n elements taken r at a time. | 5.8 Generalize
strategies for determining the number of permutations of n elements taken r at a time.
TOP: Permutations When All Objects Are Distinguishable
KEY: counting | permutation | factorial notation
28. ANS:
a) There are 15 letters and 3 positions they can be placed in.
Let U be the number of arrangements of 3 letters from UNCOPYRIGHTABLE.
There are 2730 possible 3-letter arrangements without repetition.
b) There are 14 letters and 3 positions they can be placed in.
Let A be the number of arrangements of 3 letters from AMBIDEXTROUSLY.
There are 2744 possible 3-letter arrangements with repetition.
c) Since repeating letters is allowed, there are 14 more arrangements in part b) than in part a) even
though the word is one letter shorter.
PTS: 1
DIF: Grade 12
REF: Lesson 4.3
OBJ: 5.5 Determine the number of permutations of n elements taken r at a time. | 5.8 Generalize
strategies for determining the number of permutations of n elements taken r at a time.
TOP: Permutations When All Objects Are Distinguishable
KEY: counting | permutation | factorial notation
29. ANS:
a) There are n letters and 2 positions they can be placed in.
Since n must be a positive number, Vito is thinking of a 7-letter word.
b) Answers may vary. Sample answer: SOLVING or GOLFERS
PTS: 1
DIF: Grade 12
REF: Lesson 4.3
OBJ: 5.5 Determine the number of permutations of n elements taken r at a time. | 5.8 Generalize
strategies for determining the number of permutations of n elements taken r at a time.
TOP: Permutations When All Objects Are Distinguishable
KEY: counting | permutation | factorial notation
30. ANS:
a) There are n letters and at most 2 positions they can be placed in.
Since n must be a positive number, Kira is thinking of a 10-letter word.
b) Lumberjack. Switzerland and atmospheric have 11 letters; Duplicate and trapezoid have 9
letters; Juxtaposes has a duplicate letter (S).
PTS: 1
DIF: Grade 12
REF: Lesson 4.3
OBJ: 5.5 Determine the number of permutations of n elements taken r at a time. | 5.8 Generalize
strategies for determining the number of permutations of n elements taken r at a time.
TOP: Permutations When All Objects Are Distinguishable
KEY: counting | permutation | factorial notation
31. ANS:
2+4+1=7
Let A represent the number of arrangements of 7 plates.
There are 105 different plate arrangements.
PTS: 1
DIF: Grade 12
REF: Lesson 4.4
OBJ: 5.6 Determine the number of permutations of n elements taken n at a time where some
elements are not distinct. | 5.7 Explain, using examples, the effect on the total number of
permutations of n elements when two or more elements are identical.
TOP: Permutations When Objects Are Identical
KEY: counting | permutation | factorial notation
32. ANS:
3+2+1+3=9
Let A represent the number of arrangements of 9 plates.
There are 5040 different plate arrangements.
PTS: 1
DIF: Grade 12
REF: Lesson 4.4
OBJ: 5.6 Determine the number of permutations of n elements taken n at a time where some
elements are not distinct. | 5.7 Explain, using examples, the effect on the total number of
permutations of n elements when two or more elements are identical.
TOP: Permutations When Objects Are Identical
KEY: counting | permutation | factorial notation
33. ANS:
Let M represent the number of arrangements of Ming’s 6 plates.
Let R represent the number of arrangements of Rudo’s 6 plates.
Let A represent the number of arrangements of the two stacks.
There are 900 different plate arrangements.
PTS: 1
DIF: Grade 12
REF: Lesson 4.4
OBJ: 5.6 Determine the number of permutations of n elements taken n at a time where some
elements are not distinct. | 5.7 Explain, using examples, the effect on the total number of
permutations of n elements when two or more elements are identical.
TOP: Permutations When Objects Are Identical
KEY: counting | permutation | factorial notation | Fundamental Counting Principle
34. ANS:
Let C represent the number of arrangements of Claire’s 8 plates.
Let B represent the number of arrangements of Bradley’s 10 plates.
Let A represent the number of arrangements of the two stacks.
There are 705 600 different plate arrangements.
PTS: 1
DIF: Grade 12
REF: Lesson 4.4
OBJ: 5.6 Determine the number of permutations of n elements taken n at a time where some
elements are not distinct. | 5.7 Explain, using examples, the effect on the total number of
permutations of n elements when two or more elements are identical.
TOP: Permutations When Objects Are Identical
KEY: counting | permutation | factorial notation | Fundamental Counting Principle
35. ANS:
Let A represent the number of arrangements of coins in stack 1.
Let B represent the number of arrangements of coins in stack 2.
More arrangements of coins can be made with stack 1.
PTS: 1
DIF: Grade 12
REF: Lesson 4.4
OBJ: 5.6 Determine the number of permutations of n elements taken n at a time where some
elements are not distinct. | 5.7 Explain, using examples, the effect on the total number of
permutations of n elements when two or more elements are identical.
TOP: Permutations When Objects Are Identical
KEY: counting | permutation | factorial notation
36. ANS:
Let A represent the number of arrangements of coins in stack 1.
Let B represent the number of arrangements of coins in stack 2.
More arrangements of coins can be made with stack 2.
PTS: 1
DIF: Grade 12
REF: Lesson 4.4
OBJ: 5.6 Determine the number of permutations of n elements taken n at a time where some
elements are not distinct. | 5.7 Explain, using examples, the effect on the total number of
permutations of n elements when two or more elements are identical.
TOP: Permutations When Objects Are Identical
KEY: counting | permutation | factorial notation
37. ANS:
a) Let A represent the number of arrangements of coins.
There are 900 900 possible arrangements.
b) Let B represent the number of arrangements of coins.
There are 69 300 possible arrangements with a quarter on top and a quarter on the bottom.
PTS: 1
DIF: Grade 12
REF: Lesson 4.4
OBJ: 5.6 Determine the number of permutations of n elements taken n at a time where some
elements are not distinct. | 5.7 Explain, using examples, the effect on the total number of
permutations of n elements when two or more elements are identical.
TOP: Permutations When Objects Are Identical
KEY: counting | permutation | factorial notation
38. ANS:
a) Let A represent the number of arrangements of coins.
There are 30 030 possible arrangements.
b) Let B represent the number of arrangements of coins.
There are 18 018 possible arrangements with a penny on top.
PTS: 1
DIF: Grade 12
REF: Lesson 4.4
OBJ: 5.6 Determine the number of permutations of n elements taken n at a time where some
elements are not distinct. | 5.7 Explain, using examples, the effect on the total number of
permutations of n elements when two or more elements are identical.
TOP: Permutations When Objects Are Identical
KEY: counting | permutation | factorial notation
39. ANS:
NANAIMO has 7 letters. There are 2 N’s and 2 A’s.
Let N represent the number of arrangements.
FANNY BAY has 8 letters. There are 2 A’s, 2 N’s, and 2 Y’s.
Let C represent the number of arrangements.
More arrangements can be made using the letters in FANNY BAY.
PTS: 1
DIF: Grade 12
REF: Lesson 4.4
OBJ: 5.6 Determine the number of permutations of n elements taken n at a time where some
elements are not distinct. | 5.7 Explain, using examples, the effect on the total number of
permutations of n elements when two or more elements are identical.
TOP: Permutations When Objects Are Identical
KEY: counting | permutation | factorial notation
40. ANS:
RED DEER has 7 letters. There are 2 R’s, 3 E’s, and 2 D’s.
Let R represent the number of arrangements.
REGINA has 6 different letters.
6! = 720
More arrangements can be made using the letters in REGINA.
PTS: 1
DIF: Grade 12
REF: Lesson 4.4
OBJ: 5.6 Determine the number of permutations of n elements taken n at a time where some
elements are not distinct. | 5.7 Explain, using examples, the effect on the total number of
permutations of n elements when two or more elements are identical.
TOP: Permutations When Objects Are Identical
KEY: counting | permutation | factorial notation
41. ANS:
There are 8 students and 4 positions on the committee. Order does not matter.
There are 70 different committees possible.
PTS: 1
DIF: Grade 12
REF: Lesson 4.6
OBJ: 6.1 Explain, using examples, why order is or is not important when solving problems that
involve permutations or combinations. | 6.2 Determine the number of combinations of n elements
taken r at a time. | 6.3 Generalize strategies for determining the number of combinations of n
elements taken r at a time.
TOP: Combinations
KEY: counting | combination
42. ANS:
There are 12 students and 3 positions on the committee. Order does not matter.
There are 220 different committees possible.
PTS: 1
DIF: Grade 12
REF: Lesson 4.6
OBJ: 6.1 Explain, using examples, why order is or is not important when solving problems that
involve permutations or combinations. | 6.2 Determine the number of combinations of n elements
taken r at a time. | 6.3 Generalize strategies for determining the number of combinations of n
elements taken r at a time.
TOP: Combinations
KEY: counting | combination
43. ANS:
a) There are 7 students and 4 positions on the committee. Order does not matter.
There are 35 different committees possible.
b) There are 7 students and 3 positions on the committee. Order does not matter.
There are 35 different committees possible.
c) The answers to parts a) and b) are the same because n – r = r, so the two values in the
denominator are the same.
PTS: 1
DIF: Grade 12
REF: Lesson 4.6
OBJ: 6.1 Explain, using examples, why order is or is not important when solving problems that
involve permutations or combinations. | 6.2 Determine the number of combinations of n elements
taken r at a time. | 6.3 Generalize strategies for determining the number of combinations of n
elements taken r at a time.
TOP: Combinations
KEY: counting | combination
44. ANS:
For the last room, there are 16 players and 3 beds. Order does not matter.
For the first room, there are now 13 players and 8 beds. Order does not matter.
The remaining 5 players share the middle room.
Using the Fundamental Counting Principle, the product of
the players can be assigned to the three rooms.
and
is the number of ways
There are 720 720 ways to assign the 16 players to these rooms.
PTS: 1
DIF: Grade 12
REF: Lesson 4.6
OBJ: 6.1 Explain, using examples, why order is or is not important when solving problems that
involve permutations or combinations. | 6.2 Determine the number of combinations of n elements
taken r at a time. | 6.3 Generalize strategies for determining the number of combinations of n
elements taken r at a time.
TOP: Combinations
KEY: counting | combination | Fundamental Counting Principle
45. ANS:
For the last room, there are 15 players and 4 beds. Order does not matter.
For the first room, there are now 11 players and 6 beds. Order does not matter.
The remaining 5 players share the middle room.
Using the Fundamental Counting Principle, the product of
the players can be assigned to the three rooms.
and
is the number of ways
There are 630 630 ways to assign the 15 players to these rooms.
PTS: 1
DIF: Grade 12
REF: Lesson 4.6
OBJ: 6.1 Explain, using examples, why order is or is not important when solving problems that
involve permutations or combinations. | 6.2 Determine the number of combinations of n elements
taken r at a time. | 6.3 Generalize strategies for determining the number of combinations of n
elements taken r at a time.
TOP: Combinations
KEY: counting | combination | Fundamental Counting Principle
46. ANS:
Let A represent the number of groups of 2 boys and 3 girls.
Let B represent the number of groups of 3 boys and 2 girls.
Number of groups of five = 30 030 + 23 100
Number of groups of five = 53 130
There are 53 130 groups of 5 students with those restrictions.
PTS: 1
DIF: Grade 12
REF: Lesson 4.6
OBJ: 6.1 Explain, using examples, why order is or is not important when solving problems that
involve permutations or combinations. | 6.2 Determine the number of combinations of n elements
taken r at a time. | 6.3 Generalize strategies for determining the number of combinations of n
elements taken r at a time.
TOP: Combinations
KEY: counting | combination | Fundamental Counting Principle
47. ANS:
Let A represent the number of groups of 3 boys and 3 girls.
Let B represent the number of groups of 2 boys and 4 girls.
Number of groups of six = 233 376 + 109 395
Number of groups of six = 342 771
There are 342 771 groups of 6 students with those restrictions.
PTS: 1
DIF: Grade 12
REF: Lesson 4.6
OBJ: 6.1 Explain, using examples, why order is or is not important when solving problems that
involve permutations or combinations. | 6.2 Determine the number of combinations of n elements
taken r at a time. | 6.3 Generalize strategies for determining the number of combinations of n
elements taken r at a time.
TOP: Combinations
KEY: counting | combination | Fundamental Counting Principle
48. ANS:
Let A represent the number of groups of 2 boys and 3 girls.
Let B represent the number of groups of 3 boys and 2 girls.
Let C represent the number of groups of 4 boys and 1 girls.
Let D represent the number of groups of 5 boys.
Number of groups of five = 12 240 + 3060 + 270 + 6
Number of groups of five = 15 576
There are 15 576 groups of 5 students with those restrictions.
PTS: 1
DIF: Grade 12
REF: Lesson 4.6
OBJ: 6.1 Explain, using examples, why order is or is not important when solving problems that
involve permutations or combinations. | 6.2 Determine the number of combinations of n elements
taken r at a time. | 6.3 Generalize strategies for determining the number of combinations of n
elements taken r at a time.
TOP: Combinations
KEY: counting | combination | Fundamental Counting Principle
49. ANS:
a) i)
ii)
iii)
iv)
b) The sum of the combinations of n objects is 2n.
c) 24 = 16
PTS: 1
DIF: Grade 12
REF: Lesson 4.6
OBJ: 6.1 Explain, using examples, why order is or is not important when solving problems that
involve permutations or combinations. | 6.2 Determine the number of combinations of n elements
taken r at a time. | 6.3 Generalize strategies for determining the number of combinations of n
elements taken r at a time.
TOP: Combinations
KEY: counting | combination
50. ANS:
Since the value of
increases as r gets closer to
, C is the greatest value.
PTS: 1
DIF: Grade 12
REF: Lesson 4.6
OBJ: 6.1 Explain, using examples, why order is or is not important when solving problems that
involve permutations or combinations. | 6.2 Determine the number of combinations of n elements
taken r at a time. | 6.3 Generalize strategies for determining the number of combinations of n
elements taken r at a time.
TOP: Combinations
KEY: counting | combination
51. ANS:
There are two possible orders for the groups:
• defense, goalie, forwards
• forwards, goalie, defense
Let H represent the number of arrangements:
There are 241 920 different ways the team can stand in a row.
PTS: 1
DIF: Grade 12
REF: Lesson 4.7
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 4.4 Solve a
contextual counting problem, using the fundamental counting principle, and explain the reasoning.
| 5.5 Determine the number of permutations of n elements taken r at a time. | 6.2 Determine the
number of combinations of n elements taken r at a time.
TOP: Solving Counting Problems
KEY: counting | Fundamental Counting Principle | factorial notation
52. ANS:
There are four possible orders for the groups:
• goalies, forwards, defense, coach
• goalies, defense, forwards, coach
• coach, forwards, defense, goalies
• coach, defense, forwards, goalies
Let H represent the number of arrangements:
There are 696 729 600 different ways the team can stand in a row.
PTS:
1
DIF:
Grade 12
REF: Lesson 4.7
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 4.4 Solve a
contextual counting problem, using the fundamental counting principle, and explain the reasoning.
| 5.5 Determine the number of permutations of n elements taken r at a time. | 6.2 Determine the
number of combinations of n elements taken r at a time.
TOP: Solving Counting Problems
KEY: counting | Fundamental Counting Principle | factorial notation
53. ANS:
Case 1: exactly 2 defense (and 4 not)
Case 2: exactly 1 defense (and 5 not)
Case 3: no defense
Let H represent the number of arrangements of the six players in the front row.
There are 483 840 different ways to arrange six players in the front row with at most 2 defense.
PTS: 1
DIF: Grade 12
REF: Lesson 4.7
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 4.4 Solve a
contextual counting problem, using the fundamental counting principle, and explain the reasoning.
| 5.5 Determine the number of permutations of n elements taken r at a time. | 6.2 Determine the
number of combinations of n elements taken r at a time.
TOP: Solving Counting Problems
KEY: counting | Fundamental Counting Principle | combination | permutation
54. ANS:
Case 1: exactly 1 goalie
Case 2: exactly 2 goalies
Let H represent the number of arrangements of the eight players in the front row.
There are 397 837 440 different ways to arrange eight players in the front row with at least 1
goalie.
PTS: 1
DIF: Grade 12
REF: Lesson 4.7
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 4.4 Solve a
contextual counting problem, using the fundamental counting principle, and explain the reasoning.
| 5.5 Determine the number of permutations of n elements taken r at a time. | 6.2 Determine the
number of combinations of n elements taken r at a time.
TOP: Solving Counting Problems
KEY: counting | Fundamental Counting Principle | combination | permutation
55. ANS:
5 of 12 people in water safety:
3 of the remaining 7 people in first aid:
2 of the remaining 4 people in conflict management:
2 people in astronomy:
Let C represent the number of ways to place the 12 counselors in the four courses:
There are 166 320 ways to place the counselors in the four courses.
PTS: 1
DIF: Grade 12
REF: Lesson 4.7
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 4.4 Solve a
contextual counting problem, using the fundamental counting principle, and explain the reasoning.
| 5.5 Determine the number of permutations of n elements taken r at a time. | 6.2 Determine the
number of combinations of n elements taken r at a time.
TOP: Solving Counting Problems
KEY: counting | Fundamental Counting Principle
56. ANS:
5 of 15 people in water safety:
4 of the remaining 10 people in first aid:
3 of the remaining 6 people in conflict management:
3 people in astronomy:
Let C represent the number of ways to place the 15 counselors in the four courses:
There are 12 612 600 ways to place the counselors in the four courses.
PTS: 1
DIF: Grade 12
REF: Lesson 4.7
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 4.4 Solve a
contextual counting problem, using the fundamental counting principle, and explain the reasoning.
| 5.5 Determine the number of permutations of n elements taken r at a time. | 6.2 Determine the
number of combinations of n elements taken r at a time.
TOP: Solving Counting Problems
KEY: counting | Fundamental Counting Principle
57. ANS:
12 of 20 students in the minibus:
5 of the remaining 8 students in the SUV:
3 students in the choirmaster’s car:
Let C represent the number of ways to assign rides to the 20 students:
There are 7 054 320 ways to assign rides to the 20 students.
PTS: 1
DIF: Grade 12
REF: Lesson 4.7
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 4.4 Solve a
contextual counting problem, using the fundamental counting principle, and explain the reasoning.
| 5.5 Determine the number of permutations of n elements taken r at a time. | 6.2 Determine the
number of combinations of n elements taken r at a time.
TOP: Solving Counting Problems
KEY: counting | Fundamental Counting Principle | combination
58. ANS:
16 of 25 students in the minibus:
6 of the remaining 9 students in the SUV:
3 students in the choirmaster’s car:
Let C represent the number of ways to assign rides to the 25 students:
There are 171 609 900 ways to assign rides to the 25 students.
PTS: 1
DIF: Grade 12
REF: Lesson 4.7
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 4.4 Solve a
contextual counting problem, using the fundamental counting principle, and explain the reasoning.
| 5.5 Determine the number of permutations of n elements taken r at a time. | 6.2 Determine the
number of combinations of n elements taken r at a time.
TOP: Solving Counting Problems
KEY: counting | Fundamental Counting Principle | combination
59. ANS:
Case 1: exactly 2 face cards (and 3 not)
There are 12 face cards and 40 other cards in a standard deck.
Case 2: exactly 1 face card (and 4 not)
Case 3: no face cards
Let H represent the number of hands with at most 2 face cards.
There are 2 406 768 different five-card hands that contain at most two face cards.
PTS: 1
DIF: Grade 12
REF: Lesson 4.7
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 4.4 Solve a
contextual counting problem, using the fundamental counting principle, and explain the reasoning.
| 5.5 Determine the number of permutations of n elements taken r at a time. | 6.2 Determine the
number of combinations of n elements taken r at a time.
TOP: Solving Counting Problems
KEY: counting | Fundamental Counting Principle | combination
60. ANS:
Case 1: exactly 2 face cards (and 2 not)
There are 12 face cards and 40 other cards in a standard deck.
Case 2: exactly 3 face cards (and 1 not)
Case 3: exactly 4 face cards
Let H represent the number of hands with at least 2 face cards.
There are 60 775 different four-card hands that contain at least two face cards.
PTS: 1
DIF: Grade 12
REF: Lesson 4.7
OBJ: 4.3 Identify and explain assumptions made in solving a counting problem. | 4.4 Solve a
contextual counting problem, using the fundamental counting principle, and explain the reasoning.
| 5.5 Determine the number of permutations of n elements taken r at a time. | 6.2 Determine the
number of combinations of n elements taken r at a time.
TOP: Solving Counting Problems
KEY: counting | Fundamental Counting Principle | combination