Report

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