10.3 Combinations
Objectives: Solve problems involving combinations.
Solve problems by distinguishing between
permutations and combinations.
Standards: 2.7.8 A Determine the number of
combinations and permutations for an event.
purple/green
purple/red
purple/blue
purple/grey
green/purple
green/red
green/blue
green/grey
blue/purple
blue/green
blue/red
blue/grey
grey/purple
grey/green
grey/red
grey/blue
red/purple
red/green
red/blue
red/grey
There are 10 possible
2-color combinations.
Recall that a permutation is an arrangement
of objects in a specific order.
An arrangement of objects in which order is
not important is called a combination.
1. Find the number of ways to purchase 3 different
kinds of juice from a selection of 10 different juices.
2. Find the number of ways to rent 5 comedies
from a collection of 30 comedies at a video store.
6
7 9 13
30!
n Cr  30 C5 
5!(30  5)!

30!
30  29  28  27  26  25!

5!25!
5  4  3  2  25!
 6  29  7  9  13  142506
3. Find the number of combination of 9 objects
taken 7 at a time.
9!
 36
9 C7 
7!2!
c) How many ways are there to give 3 honorable
mentions awards to a group of 8 entrants in a
contest?
8C3
= 56
d) How many ways are there to award 1st, 2nd,
3rd prize to a group of 8 entrants in a contest?
8P3
= 336
e) How many ways are there to choose a committee
of 2 people from a group of 7 people?
7C2
= 21
f) How many ways are there to choose a chairperson
and a co-chairperson from a group of 7 people?
7P2
= 42
Consider CD’s, cassettes, and videotapes separately, and apply the
fundamental counting principle.
a) How many different ways are there to purchase 3 CDs, 4
cassettes, and 2 videotapes if there are 3 CD titles, 6
cassette titles, and 4 videotape titles from which to choose?
3C3
x 6C4 x 4C2 = 90
b) Terry is buying paperback books to read while on
vacation. How many different ways are there for Terry to
purchase 3 novels and 2 non-fiction books if there are 10
novels and 6 nonfiction books to choose from?
10C3
x 6C2 = 1800
Using Combinations and Probability
4) In a recent survey of 25 voters, 17 favor a new city regulation and 8
oppose it. Find the probability that in a random sample of 6
respondents from this survey, exactly 2 favor the proposed regulation
and 4 oppose it.
First, find the number of outcomes in the event. Use the Fundamental
Counting Principle.
17
C2 
8
C4
Choose 4 of the 8 who oppose.
Choose 2 of the 17 in favor.
Next, find the numbers of outcomes in the sample space.
25
Finally, find the probability.
C6
Choose 6 from the 25 respondents.
number of outcomes in event A

number of outcomes in the sample space
17
C2  8 C4
 0.05
25 C6
Thus, the probability of selecting exactly 2 people in favor and 4 people opposed in
a randomly selected group of 6 is about 5%
number of outcomes in event A

number of outcomes in the sample space
17
C2  8 C4
 0.05
25 C6
Thus, the probability of selecting exactly 2 people in favor and 4 people opposed in
a randomly selected group of 6 is about 5%
5) In a recent survey of 30 students, 25 students favored an earlier opening
time for the school cafeteria and 5 opposed it. Find the probability that
in a random sample of 8 respondents from this survey, exactly 6 favored
the earlier opening time and exactly 2 opposed it.
First, find the number of outcomes in the event. Use the Fundamental
Counting Principle.
25
C6 
5
C2
Choose 2 of the 5 who oppose.
Choose 6 of the 25 in favor.
Next, find the numbers of outcomes in the sample space.
30
Finally, find the probability.
C8
Choose 8 from the 30 respondents.
number of outcomes in event A

number of outcomes in the sample space
25
C6  5 C2
 0.30
30 C8
Thus, the probability of selecting exactly 6 students in favor and 2 students
opposed in a randomly selected group of 8 is about 30%
HOMEWORK
Practice 10.3
Quiz TOMORROW on 10.1 – 10.3