Transcript 10.3 - Souderton Math
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 red/purple red/green red/blue red/grey blue/purple blue/green blue/red blue/grey grey/purple grey/green grey/red grey/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
n C r
from a collection of 30 comedies at a video store.
30
C
5 30!
30!
5!25!
6 7 9 13
3. Find the number of combination of 9 objects taken 7 at a time.
9
C
7 9!
7!2!
36
c) How many ways are there to give 3 honorable mentions awards to a group of 8 entrants in a contest?
8
C
3
= 56
d) How many ways are there to award 1st, 2nd, 3rd prize to a group of 8 entrants in a contest?
8
P
3
= 336
e) How many ways are there to choose a committee of 2 people from a group of 7 people?
7
C
2
= 21
f) How many ways are there to choose a chairperson and a co-chairperson from a group of 7 people?
7
P
2
= 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?
3 C 3 x 6 C 4 x 4 C 2 = 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?
10 C 3 x 6 C 2 = 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
C
2 8
C
4 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
C
6 Choose 6 from the 25 respondents.
Finally, find the probability.
number of outcomes in event A number of outcomes in the sample space 17
C
2 25
C
6 8
C
4 0.05
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
C
2 25
C
6 8
C
4 0.05
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
C
6 5
C
2 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
C
8 Choose 8 from the 30 respondents.
Finally, find the probability.
number of outcomes in event A number of outcomes in the sample space 25
C
6 30
C
8 5
C
2 0.30
Thus, the probability of selecting exactly 6 students in favor and 2 students opposed in a randomly selected group of 8 is about 30%