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