Name:
Date:
Algebra 2
Counting (Day 1): Multiplication, Factorials, Permutations
Objective: You will solve counting problems using the Multiplication Principle, factorials,
and permutations.
Today’s lesson begins a unit on counting problems (counting the number of different ways that
things can be chosen, selected, arranged, etc.). Initially we’ll be reviewing what you learned about
counting in 9th grade, but eventually we will take the topic further than before.
The Multiplication Principle
Here is a simple but versatile counting strategy called the Multiplication Principle:
When you have two or more choices to make, the total number of possible ways
of choosing is found by multiplying the numbers of options for each choice.
Examples
Q. If someone has 5 shirts and 3 jeans, how many “outfits” (shirt and jeans) can they make?
A. The number of possible outfits is 5 · 3 = 15.
Q. From a list of 10 songs, you are asked to pick your 1st, 2nd, and 3rd favorite songs.
How many different ways of choosing are there?
A. You have 10 choices for the 1st song, 9 choices for the 2nd song (because you can’t choose
the same song again), and 8 choices for the 3rd song (do you know why?).
So, the total number of ways of choosing is 10 · 9 · 8 = 720.
Problems
1. The Class Council is selling T-shirts that come in 3 colors (yellow, blue, white) and
4 sizes (small, medium, large, extra large). How many different kinds of T-shirts are available?
2. A card game uses a special deck where each card has a number and a color.
The numbers range from 1 to 12. The colors are red, orange, yellow, green, and blue.
For example, there is a “5 blue” card. If the deck has one card with each of the possible
pairings of number and color, how many cards are there in the deck?
3. The table d’hôte (complete meal) at a French restaurant consists of a salad, an entree, and
a dessert. If there are 2 salad choices, 5 entree choices, and 3 dessert choices, how many
different meals are there?
Name:
Date:
Algebra 2
4. Suppose you flip a coin 4 times. Each time, the coin is either heads (H) or tails (T).
An example of a sequence of 4 coin flips would be: HTTH.
a. List all of the different sequences of 4 coin flips. (The best way to find all of them and be
sure you have a complete list, is to organize your list in some specific order, rather than
using a random order.)
b. Answer these questions to count the number of flip sequences for 4 coins.
o How many possible outcomes are there for the first flip?
____
o How many possible outcomes are there for the second flip?
____
o How many possible outcomes are there for the third flip?
____
o How many possible outcomes are there for the fourth flip?
____
o Using the Multiplication Principle, how many sequences of flips are there? _______
c. Make sure that your counting answer to part b agrees with the length of your list in part a.
5. If a coin is flipped 6 times, how many different sequences of heads and tails are there?
6. If a coin is flipped 20 times, how many different sequences of heads and tails are there?
(Find a way to get the answer that’s faster than multiplying 20 numbers.)
7. If a coin is flipped N times, how many different sequences of heads and tails are there?
Name:
Date:
Algebra 2
Counting shortcut: using an exponent
Sometimes a multiplication counting problem involves multiplying the same number repeatedly.
If so, you can calculate the answer using an exponent. For example, the answers to the last three
problems were 26, 220, and 2n.
More problems
8. a. How many different ways could you answer a set of 10 multiple-choice questions,
where each question has answer choices {A, B, C, D, E}?
b. How many different ways could you answer a set of n multiple-choice questions,
where each question has answer choices {A, B, C, D, E}?
9. A series of Massachusetts license plates has 4 numbers followed by 2 letters.
Here are some examples of license plates: 1234 AB, 5555 XY, 7070 EE.
Answer these questions to figure out how many different license plates are possible.
o How many possibilities are there for the first number?
Hint: The possible digits are 0 through 9.
____
o How many possibilities are there for the second number?
____
o How many possibilities are there for the third number?
____
o How many possibilities are there for the fourth number?
____
o How many possibilities are there for the first letter?
____
o How many possibilities are there for the second letter?
____
o Now, using the Multiplication Principle, how many license plates are there? _________
10. A series of New Hampshire license plates has 3 letters followed by 3 numbers.
Here are some examples: ABC 123, SPY 007, ZZZ 789.
How many license plates of this kind are possible?
Name:
Date:
Algebra 2
11. At a bank, every ATM card has a number password called a PIN.
Here are some examples of PIN’s: 7486, 0122, 9999.
Answer these questions to figure out how many different PIN’s are possible.
o How many possibilities are there for the first digit of the PIN?
____
o How many possibilities are there for the second digit of the PIN? ____
o How many possibilities are there for the third digit of the PIN?
____
o How many possibilities are there for the fourth digit of the PIN? ____
o Now, using the Multiplication Principle, how many sequences are there? _________
12. A web site requires passwords consisting of exactly 5 capital letters.
Here are some examples of passwords: ABCDE, LEXMA, HELLO.
How many different passwords are possible?
Factorials
Try this problem which exemplifies a counting concept called factorial.
13. There are 5 baseball teams in the American League Eastern Division. Some of you will be
unhappy that I bring this up, but the teams’ standings for the 2010 season were:
1st place:
2nd place:
3rd place:
4th place:
5th place:
Tampa Bay
New York
Boston
Toronto
Baltimore
The teams just started the 2011 season. How many different standings are possible for this
new season? Figure this out by answering the following series of questions.
[Disregard the possibility of teams having a tie. Assume that there will definitely be one team
in 1st place, one team in 2nd place, and so on.]
o How many possibilities are there for the 1st place team? ____
o After choosing the 1st place team,
how many possibilities are there for the 2nd place team? ____
o After choosing the 1st and 2nd place teams,
how many possibilities are there for the 3rd place team? ____
o After choosing the 1st, 2nd, and 3rd place teams,
how many possibilities are there for the 4th place team? ____
o After choosing the 1st, 2nd, 3rd, and 4th place teams,
how many possibilities are there for the 5th place team? ____
o Now, using the Multiplication Principle, how many different standings are possible?
Name:
Date:
Algebra 2
Counting shortcut: using a factorial
Sometimes a counting problem involves multiplying all of the whole numbers from a certain
number down to 1.
For example, the answer to problem 13 is 5·4·3·2·1 = 120. This answer is called “5 factorial” and
has the symbol “5!”
In general, any problem asking how many ways n objects can be arranged in different orders, will
have the answer n! , which is the product of all the whole numbers from n down to 1.
Factorial problems
14. a. How many ways can 11 objects be arranged in different orders? Answer this question by
calculating 11!
b. How many ways can the 26 letters of the alphabet be arranged in different orders?
Answer this question by calculating 26!
15. The digits 1234567 can be rearranged in many different orders.
Here are some examples: 7243165, 5162743, 6512374.
How many different digit orders are possible?
a. Write the answer to this question using a factorial symbol.
b. Write the answer to this question as a multiplication.
c. Write the answer to this question as an ordinary whole number.
Name:
Date:
Algebra 2
Permutations
Try this problem which exemplifies a counting concept called permutation.
16. A softball coach has 15 players on her roster. She needs to choose 9 players to play in a
game: someone to bat 1st, someone to bat 2nd, someone to bat 3rd, and so on through 9th.
How many different ways can the coach make her choices? Figure this out by answering the
following series of questions.
o How many possibilities are there for the 1st player chosen?
____
o How many possibilities are there for the 2nd player chosen? ____
o How many possibilities are there for the 3rd player chosen? ____
o How many possibilities are there for the 4th player chosen? ____
o How many possibilities are there for the 5th player chosen? ____
o How many possibilities are there for the 6th player chosen? ____
o How many possibilities are there for the 7th player chosen? ____
o How many possibilities are there for the 8th player chosen? ____
o How many possibilities are there for the 9th player chosen? ____
o Now, using the Multiplication Principle, how many different ways can the coach make
her choices?
Name:
Date:
Algebra 2
Counting shortcut: using a permutation
Sometimes a counting problem involves multiplying all of the whole numbers from a certain
number down to 1.
For example, the answer to problem 16 is 15·14·13·12·11·10·9·8·7 = 1,816,214,400.
This answer is called “15 permutation 9” and has the symbol “ 15P9 .”
In general, any problem asking how many ways k out of n objects can be chosen with an order
will have the answer nPk , which is the product of the first k whole numbers counting downward
from n.
Permutation problems
17. a. How many ways can 4 out of 12 objects be chosen with an order? Answer this question
by calculating 12P4.
b. How many ways can 3 different letters of the alphabet be chosen with an order? Answer
this question by calculating 26P3.
18. Suppose there are 100 students with projects entered in a science fair. Of them, 5 students
must be chosen to be win 1st, 2nd, 3rd, 4th, and 5th prize. How many different ways could
the winners be chosen?
a. Write the answer to this question using a permutation symbol.
b. Write the answer to this question as a multiplication.
c. Write the answer to this question as an ordinary whole number.
Name:
Date:
Algebra 2
Factorials and permutations together
Directions for problems 19–26: For each counting problem, first identify the answer as a
factorial or as a permutation (symbol using either ! or P), then write the answer as a
multiplication, and finally give the answer as a plain number. Here are two examples of how
much work to show:
26P5
= 26 · 25 · 24 · 23 · 22 = 7,893,600.
9! = 9 · 8 · 7 · 6 · 5 · 4 · 3 · 2 · 1 = 362,880.
19. A survey asks you to give your 1st, 2nd, and 3rd favorite months of the year. How many
different ways can you answer this survey?
20. There are 12 students in a class. Each student has to take a turn giving an oral report to the
class. In how many different orders could the students give their reports?
21. There are 25 players on a Major League baseball team. The manager has to pick 9 of them
to play the game, and put them in a batting order (1st through 9th). How many different ways
can these choices be made?
22. The 6 Lexpress buses (numbered 1 through 6) are lined up at a bus stop. In how many
different orders could the buses be lined up?
Name:
Date:
Algebra 2
23. A club with 20 members needs to choose four different people to be the president,
vice-president, secretary, and treasurer. How many ways can this be done? Hint: Think of
the jobs as being like 1st, 2nd, 3rd, and 4th prize, then use the same method as in problem 18.
24. How many letter sequences are there consisting of three different capital letters?
(Examples of the sequences you’re counting: ALG, QXV; but not KAK.)
25. A store has to hire a manager and an assistant manager. Ten people are interviewed for
the jobs. How many different ways can the hiring decisions be made?
26. A family of four needs to assign a chore to each person: one person will wash the dishes,
another person will take out the trash, another person will clean the bathroom, and the last
person will mow the lawn. How many different ways can the chores be assigned?
Check your answers
1. 3 · 4 = 12
2.
12 · 5 = 60
Name:
Date:
Algebra 2
3. 2 · 5 · 3 = 30
4b. 2 · 2 · 2 · 2 = 16
5. 2 · 2 · 2 · 2 · 2 · 2 = 26 = 64
6. 220 = 1,048,576
7. 2n
8a. 510 = 9,765,625
9. 10 · 10 · 10 · 10 · 26 · 26 = 6,760,000
10. 26 · 26 · 26 · 10 · 10 · 10 · = 17,576,000
11. 10 · 10 · 10 · 10 = 10,000
12. 26 · 26 · 26 · 26 · 26 = 11,881,376
8b. 5n
13. 5 · 4 · 3 · 2 · 1 = 120
14a. 11! = 11 · 10 · 9 · 8 · 7 · 6 · 5 · 4 · 3 · 2 · 1 = 39,916,800
14b. 26! = 26 · 25 · 24 · ··· · 3 · 2 · 1 ≈ 4.032915 · 1026
15. 7! = 7 · 6 · 5 · 4 · 3 · 2 · 1 = 5,040
16. 15·14·13·12·11·10·9·8·7 = 1,816,214,400.
17a. 12P4 = 12·11·10·9 = 11,880
17b. 26P3 = 26 · 25 · 24 = 15,600
18.
100P5
= 9,034,502,400
19. 12P3 = 1,320
20. 12! = 479,001,600
21. 25P9 ≈ 741,354,768,000
22. 6! = 720
23. 20P4 = 116,280
24.
25. 10P2 = 90
26. 4! = 24
26P3
= 15,600