Unit 10 Review
Advanced Algebra B
Reference: McDougal Littell: §10.1-10.6. Pages 734-736. See Problems 1-29
Fundamental Counting Principle.
1. How many odd 3-digit positive integers can be written using the digits 2, 3, 4, 5 and 6 (digits can repeat)?
5•5• 2 = 50
2. A student council has 5 seniors, 4 juniors, 3 sophomores and 2 freshmen as members. In how many ways can a
4-member council committee be formed that includes one member from each class? 5•4•3•2 = 120
3. How many positive, odd, natural numbers less than 60,000 can be made from the digits 0, 1, 4, 5, 6, 9 (digits can
repeat)? 3•6•6•6•3+5•6•6•3+5•6•3+5•3+3 = 2592
4. How many ways are there to select three cards from a standard deck of cards if:
(a) The cards are replaced after each draw. 52•52•52 = 140,608
(b) The cards are not replaced after each draw. 52•51•50 = 132,600 (if the order of the draws matters)
5. How many 7 digit phone numbers can be created if the first digit must be 8, the second must be 5, and the third must
be 2 or 3? 1•1•2•10•10•10•10 = 20,000
6. Suppose you have totally forgotten the combination to your locker. There are three numbers in the combination, and
you’re sure each number is different. The numbers on the lock’s dial range from 0 to 35. If you test one combination
every 12 seconds, how long will it take you to test all possible combinations? 36•35•34 = 42,840•12 = 514,080
seconds = 5.95 days!
7. How many 3-letter code words can be formed if at least one of the letters has to be one of the vowels a, e, i, o, or u?
Ignore.
Permutations.
Evaluate.
8. 6!
9. (3!)(2!)
10.
6•5•4•3•2•1 = 720
(3•2•1)•(2•1) = 12
8!
4!
8  7  6  5  4  3  2 1
4  3  2 1
(n - 1)!
(n - 1)(n - 2)(n - 3)(n - 4)...
1
12.
=

( n  1)! ( n  1) n(n - 1)(n - 2)(n - 3)(n - 4)... ( n  1)n
11. n(n – 1)!
 1680
n(n  1)( n  2)( n  3)...(3)( 2)(1) or
By definition: n!
13. In how many ways can 6 books be arranged on a shelf?
6! = 6 P6  720
14. In how many ways can 3 cards from a deck of 52 cards be arranged in a row face up?
15. How many different ways can the letters of the work BEEKEEPER be arranged?
9!
5!
52•51•50 =
52
P3  132,600
 3,024
16. In how many ways can 3 red, 4 blue and 2 green pens be distributed to 9 students seated in a row if each student
receives only one pen?
9!
3!  4!  2!
 1,260
Combinations.
Expand.
17. (x + 2y)5
18. (3 – 2a)4
x 5  10 x 4 y  40 x 3 y 2  80 x 2 y 3  80 xy 4  32 y 5
81  216a  216a 2  96a 3  16a 4
19. In how many ways can a committee of 6 people be chosen from 5 teachers and 4 students if:
(a) all are equally eligible? 9 C 6  84
(b) the committee must include 3 teachers and 3 students? 5 C3  4 C3  40
20. How many 5 card hands can be dealt from a standard deck of 52 cards that include:
(a) 3 hearts and 2 clubs? 13 C3 13 C 2  22,308
(b) At least 4 spades?
13
C4 39 C1 13 C5  29,172
21. A school club has 15 boys and 16 girls as members. How many different 6-person committees can be formed if:
(a) equal numbers of boys and girls are on the committee? 15 C3 16 C3  254,800
(b) 4 boys are on the committee?
15
C4 16 C2  163,800
(c) at least 4 boys are on the committee? 15 C 4 16 C 2 15 C5 16 C1 15 C6  216,853
Probability.
22. There are 12 tulip bulbs in a package. Nine will yield yellow tulips and 3 will yield red tulips. If two tulips are
selected at random, find the probability that:
C2
1

 .045
22
12 C 2
C
9
9 C1
(b) One of each color will be chosen.
 3 1 
 .409
22
12 C 2
12 C 2
1
9

 .45
(c) At least one tulip will be red.
22 22
(a) Two red tulips will be chosen.
3
23. A letter is chosen at random from the letters in the word TRIANGLE. Find the probability of each event.
(a) It is a vowel.
3
8
(b) It is from the first half of the alphabet.
(c) It is a consonant or from the first half of the alphabet.
5
8
5 5 2
  1
8 8 8
24. A bag contains 2 red, 4 yellow and 6 blue marbles. Three marbles are chosen at random. Find the probability of each
event.
C3
1
(b) 2 blue are chosen.

 .018
55
12 C 3
C  C
5
6 C 2  6 C1
(c) 2 blue or 2 red are chosen.
 2 2 10 1 
 .45
22
12 C3
12 C3
(a) 3 yellow are chosen.
4
6
C 2  6 C1
9

 .409
22
12 C 3
25. The marbles in problem 25 are chosen one at a time without replacement. Find the probability of each event.
C1 4 C1 6 C1
2



 .036
55
12 C1
11 C1
10 C1
C
C
3
4 C1
(b) A yellow, then a blue, then a yellow is chosen.
 6 1  3 1 
 .054
55
12 C1
11 C1
10 C1
(a) A red, then a yellow then a blue is chosen.
2
26. If a six volume set of books is placed on a shelf at random, what is the probability that the books will be arranged in
either the correct or reverse order?
2
1

6! 360
27. Of 200 students at a school, 58 play football, 40 play basketball, and 18 play football or basketball but not both. Find
the probability that a randomly selected student plays both football and basketball.
58
40
18
80
2




200 200 200 200 5
Binomial Distribution.
28. A study found that 9% of people cite fishing as their favorite leisure activity. 8 people are randomly chosen and
asked about their favorite leisure activity.
(a) Is this experiment a binomial experiment? Explain. Yes. Fishing/Not Fishing.
(b) Make a histogram of the probability distribution of those favoring fishing as a leisure activity.
(c) Find the probability that exactly 3 people surveyed will cite fishing as their favorite activity. P(3) = .03
(d) What is the probability that at least 6 people will cite fishing as their favorite activity? P(≥6) = .0000126543