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/greenpurple/redpurple/bluepurple/grey

green/purplegreen/redgreen/bluegreen/grey

red/purplered/greenred/bluered/grey

blue/purpleblue/greenblue/redblue/grey

grey/purplegrey/greengrey/redgrey/blue

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.

3. Find the number of combination of 9 objects taken 7 at a time.

30 530!

5!(30 5)!n rC C

30! 30 29 28 27 26 25!5!25! 5 4 3 2 25!

6

9 79! 36

7!2!C

7 9 13

6 29 7 9 13 142506

c) How many ways are there to give 3 honorable mentions awards to a group of 8 entrants in a contest?

d)How many ways are there to award 1st, 2nd, 3rd prize to a group of 8 entrants in a contest?

8C3 = 56

8P3 = 336

e) How many ways are there to choose a committee of 2 people from a group of 7 people?

f) How many ways are there to choose a chairperson and a co-chairperson from a group of 7 people?

7C2 = 21

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?

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?

3C3 x 6C4 x 4C2 = 90

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 2 8 4 C C

Choose 2 of the 17 in favor. Choose 4 of the 8 who oppose.

Next, find the numbers of outcomes in the sample space.

25 6C Choose 6 from the 25 respondents.Finally, find the probability.

17 2 8 4

25 6

number of outcomes in event A 0.05number of outcomes in the sample space

C CC

Thus, the probability of selecting exactly 2 people in favor and 4 people opposed in a randomly selected group of 6 is about 5%

17 2 8 4

25 6

number of outcomes in event A 0.05number of outcomes in the sample space

C CC

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 6 5 2 C C

Choose 6 of the 25 in favor. Choose 2 of the 5 who oppose.

Next, find the numbers of outcomes in the sample space.

30 8C Choose 8 from the 30 respondents.Finally, find the probability.

25 6 5 2

30 8

number of outcomes in event A 0.30number of outcomes in the sample space

C CC

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