LESSON
Page 1 of 6
11.7
Combinations
Now
BEFORE
Vocabulary
You used permutations
to count possibilities.
WHY?
You’ll use combinations to
count possibilities.
So you can count possible
yearbook designs, as in Ex. 9.
combination, p. 620
Basketball A basketball league has
5 teams. Two of the teams are chosen
to play one another. How many different
possible matchups are there?
In Lesson 11.6, you learned that a
permutation is an arrangement in which
order is important. A combination is a
selection of objects where the order in
which the objects are chosen is not
important. In the problem above, suppose
team A and team B are the two teams
selected to play one another. It does not
matter if team A is chosen first or if
team B is chosen first. The same game
will be played.
Example 1
Listing Combinations
List and count the different possible matchups of the basketball teams
described above.
Solution
Use the letters A, B, C, D, and E to represent the five teams. List all
possible matchups. Then cross out any duplicates that represent the
same matchup.
AB
AC
AD
AE
BA
BC
BD
BE
CA
CB
CD
CE
DA
DB
DC
DE
EA
EB
EC
ED
DE and ED are duplicates, because
they represent the same matchup.
Answer There are 10 different matchups.
Checkpoint
1. You want to buy 4 different CDs. You can afford to buy only 3 of
them. How many combinations of CDs can you buy?
620
Chapter 11
Data Analysis and Probability
Page 2 of 6
Study Strategy
For every combination of
2 teams, there are 2!, or 2,
permutations of the chosen
teams. So, 5P2 2! p 5C2, or
P
2!
5 2
.
C 5 2
Relating Combinations and Permutations Before matchups are crossed
out, the list in Example 1 shows the permutations of 5 teams chosen
2 at a time, or 5P2. After the duplicate matchups are crossed out, the list
shows the number of combinations of 5 teams chosen 2 at a time. This is
P
5 2
, which suggests the
written as 5C 2. Notice that in this situation 5C 2 2!
following result.
Combinations
Words To find the number of combinations of n objects taken
r at a time, divide the number of permutations of n objects taken
r at a time by r!.
P
9 5
Numbers 9C5 5!
Example 2
P
n r
Algebra nCr r!
Counting Combinations
Book Reports You need to write 4 book reports for your English class.
Your teacher gives the class a list of 7 books from which to choose.
How many different groups of 4 books can you choose from the list?
Solution
The order in which you choose the books is not important. So, to find
the number of different ways to choose 4 books from 7, find 7C4.
P
4!
7 4
C 7 4
Use combinations formula.
7p6p5p4
4p3p2p1
2
1
Write 7 P4 and 4! as products.
1
7p6p5p4
4p3p2p1
Divide out common factors.
35
Simplify.
1
1
1
Answer There are 35 different combinations of books.
Checkpoint
Find the number of combinations.
2. 6C 3
3. 8C 5
4.
C
10 2
Using a Calculator Many calculators can
evaluate combinations. The solution for
Example 2 is shown. You may find it helpful
to use a calculator when evaluating
combinations that involve large numbers.
Lesson 11.7
5. 4C4
7 nCr 4
35
Combinations
621
Page 3 of 6
You can determine whether a problem requires permutations or
combinations by deciding if order is important.
Example 3
Choosing Between Permutations and Combinations
Tell whether the possibilities can be counted using permutations or
combinations.
a. There are 30 dogs in a dog show. Blue ribbons are awarded to the
top 3 dogs. How many different groups of dogs can receive blue
ribbons?
b. There are 30 runners in a cross country race. How many different
groups of runners can finish first, second, and third?
Solution
a. The order in which the 3 blue ribbons are awarded does not
matter. So, the possibilities can be counted using combinations.
b. Because the runners can finish first, second, or third, order is
important. So, the possibilities can be counted using
permutations.
Example 4
Finding a Probability Using Combinations
Marbles You have 10 marbles in a bag. Each marble is a different
color. You draw 3 marbles at random. Find the probability that you
draw a red, a blue, and a green marble.
Solution
The order in which the marbles are drawn is not important. Find 10C3.
P
3!
10 3
C 10 3
Use combinations formula.
10 p 9 p 8
3p2p1
3
Write
P and 3! as products.
10 3
4
10 p 9 p 8
3p2p1
Divide out common factors.
120
Simplify.
1
1
Answer There are 120 different combinations of 3 marbles you can
draw. Only one of the combinations includes a red, a blue, and a green
1
120
marble. So, the probability is .
Checkpoint
6. You are choosing 10 of 30 friends to invite to a party. Can you count
the possibilities using permutations or combinations?
7. You have 12 marbles in a bag. Each marble is a different color. You
draw 2 marbles at random. What is the probability that you draw a
red marble and a blue marble?
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Chapter 11
Data Analysis and Probability
Page 4 of 6
11.7
Exercises
INTERNET
More Practice, p. 813
CLASSZONE.COM
eWorkbook Plus
Guided Practice
Vocabulary Check
1. Copy and complete: You can write the number of combinations of
12 objects taken 3 at a time as _?_.
2. Explain when to use a permutation and when to use a combination
when choosing r objects from n objects.
Skill Check
Find the number of combinations.
3. 2C 1
4. 5C4
5. 3C 3
6. 6C4
7. Track and Field A gym coach selects a team of 4 athletes from
14 athletes to represent the school at a track and field meet. Tell
whether the number of teams the coach can select can be counted
using permutations or combinations. Then find the number of teams.
Guided Problem Solving
8. Concert A radio station takes the names of the first 20 listeners who
call in after hearing a certain song. The station will randomly select 3 of
the callers to win tickets to a concert. If 3 friends are among the first
20 callers, what is the probability that the 3 friends will win tickets?
1
How many possible combinations of 3 callers can be selected to
win tickets?
2
How many possible combinations of 3 ticket winners include all
3 of the friends?
3
Find the probability that the 3 friends win tickets.
Practice and Problem Solving
Homework Help
Example
1
2
3
4
Exercises
9, 10
11–22
23–26, 28
27, 31
9. Yearbook The school yearbook photographer took one photo of the
school during each of the 4 seasons and wants to choose 3 of the
photos for the cover. How many sets of 3 photos are there? Make a list
of all the possible combinations.
10. Club Meetings A club is scheduling a week of meetings to plan for its
annual fundraiser. The club wants to meet on 3 of the evenings next
week, from Monday through Friday. How many different schedules can
the club choose? Make a list of all the possible combinations.
Online Resources
CLASSZONE.COM
• More Examples
• eTutorial Plus
Find the number of combinations.
11. 3C 2
12. 4C 1
13. 7C 3
14. 6C 5
15. 4C 3
16. 9C 8
17. 8C 3
18.
C
21. 7C 7
22.
19.
C
12 3
20.
15 6
Lesson 11.7
C
11 7
C
10 0
Combinations
623
Page 5 of 6
In Exercises 23–25, tell whether the possibilities can be counted using
permutations or combinations. Then answer the question.
23. A survey asks people to rank basketball, baseball, tennis, soccer, and
football according to how much they enjoy watching each sport. How
many possible responses are there?
24. A history test lists the names of 5 presidents, and each student is to
choose two of them to compare in an essay. How many different
pairings are possible?
25. A subway car has 8 empty seats. At one stop, 5 people enter the car and
no one gets off. How many ways can the 5 people arrange themselves
in the 8 empty seats if each person takes one seat?
26. Talent Show At an upcoming talent show, you plan to play 3 songs
on the piano. There are 6 songs you know well enough to perform.
a. How many different groups of 3 songs can you choose to play?
b. Once you have chosen 3 songs, in how many ways can you play
them at the talent show?
27. Chocolates A box of chocolates contains 10 pieces of chocolate, and
4 of the pieces have cream in the center. Suppose you randomly select
4 pieces of chocolate. Find the probability that you select all 4 of the
pieces that have cream in the center.
28. Tiles A designer is making a sample design that will use 3 different
kinds of tiles. The designer has 9 different kinds of tiles from which
to choose.
a. How many possible combinations of tiles can the designer choose?
b. The designer will create a sample design by placing 3 tiles side by
side. How many different sample designs can the designer make
from the 3 chosen tiles?
29.
Writing
Explain why 5C4 5C1. Find another example of two different
combinations that are equal.
30. Critical Thinking Which is greater, 4C2 or 4P2? Explain why.
31. Raffle In a raffle, 2 of 50 tickets are randomly selected to be the
winning tickets.
a. You have 2 raffle tickets. Find the probability that you are holding
both winning tickets.
b. Does buying twice as many tickets double the probability that you
are holding both winning tickets? Explain.
32. Pizza You and your friends are choosing toppings for a pizza. There are
4 meat toppings and 6 vegetable toppings from which to choose.
a. How many pizzas having 1 meat topping and 2 different vegetable
toppings can you choose?
b. How many pizzas having 2 different meat toppings and 1 vegetable
topping can you choose?
c. How many pizzas having 3 different toppings can you choose?
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Chapter 11
Data Analysis and Probability
Page 6 of 6
33. Break into Parts How many ways can you choose three of the numbers
Review Help
3, 4, 6, 8, and 9 so that at least one of the numbers is greater than 6?
Explain.
34. Critical Thinking You have 100 different coins. Are there more
To review the strategy break
into parts, see p. 802.
combinations of 99 of the coins than there are of 100 coins? Explain.
35. Challenge The numbers 1 through 5 are written on separate slips of
paper and placed in a hat, and 3 of the slips are drawn randomly. What
is the probability that one of the slips that are drawn shows a 4?
Mixed Review
36. Roads There are 3 roads from town A to town B. There are 4 roads from
town B to town C. How many different ways are there to get from
town A to town B to town C? (Lesson 6.8)
In Exercises 37 and 38, make a box-and-whisker plot of the data.
(Lesson 11.2)
37. Basketball game scores: 98, 104, 93, 96, 122, 106, 102, 107
38. Temperatures (in degrees Fahrenheit): 30, 35, 38, 28, 41, 30, 40, 37, 40,
36, 38
39. Movies You rent 6 movies. In how many different ways can you watch
all 6 of the movies? (Lesson 11.6)
Standardized Test
Practice
40. Multiple Choice What is the value of 8C4?
A. 56
B. 70
C. 1680
D. 40,296
41. Multiple Choice Two cars are available for 7 students to take to a
high school football game. Car A holds 4 students, and car B holds
3 students. How many different groups of 3 students can be formed
to ride in car B?
F. 4
G. 24
H. 35
I. 210
Playoff Series
In many playoff series, the team that wins a majority of the games
is the winner. In a best-of-five series, the first team to win 3 games
is the winner of the series. One way to win a best-of-five series is to
win the first 3 games. Another way is to lose the first 2 games, then
win the next 3.
How many different ways are there to win a
best-of-three series? a best-of-five series?
a best-of-seven series?
Lesson 11.7
Combinations
625