Copyright © 2016, 2013, and 2010, Pearson Education, Inc. 9 Chapter Probability Copyright © 2016, 2013, and 2010, Pearson Education, Inc.

9-4 Permutations and Combinations in Probability Students will be able to understand and explain: Permutations; Permutations of like objects; Combinations; and Use of permutations and combinations in probability problems.

Permutations of Unlike Objects An arrangement of things in a definite order with no repetitions Fundamental Counting Principle If an event M can occur in m ways and, after M has occurred, event N can occur in n ways, then event M followed by event N can occur in m · n ways.

Definition n factorial Compute: 0! 1! 2! 3! 4! 5! 6! 7!

Permutations of Objects in a Set The number of permutations of r objects chosen from a set of n objects, where 0  r  n is denoted by and is given by

Example a. A baseball team has nine players. Find the number of ways the manager can arrange the batting order. b. Find the number of ways of choosing three initials from the alphabet if none of the letters can be repeated.

Permutations Involving Like Objects If there are n objects, of which r1 are alike, r2 are alike, and so on through rk, then the number of different arrangements of all n objects, where alike objects are indistinguishable, is equal to

Example Find the number of rearrangements of the letters in each of the following words: a. bubble b. statistics

Combinations Combination An arrangement of objects in which the order makes no difference

Combinations To find the number of combinations possible in a counting problem, find the number of permutations and then divide by the number of ways in which each choice can be arranged.

Example The Library of Science Book Club offers 3 free books from a list of 42. How many possible combinations are there? Order is not important, so this is a combination problem. There are 42 · 41 · 40 ways to choose the free books. The three circled numbers can be arranged in 3 · 2 · 1 ways.

Example (continued)

Example At the beginning of the first semester of a mathematics class for elementary school teachers, each of the class’s 25 students shook hands with each of the other students exactly once. How many handshakes took place? Since the handshake between persons A and B is the same as that between persons B and A, this is a problem of choosing combinations of 25 people 2 at a time.

Example Given a class of 12 girls and 10 boys, answer each of the following: a. In how many ways can a committee of 5 consisting of 3 girls and 2 boys be chosen? The girls can be chosen in 12C3 ways. The boys can be chosen in 10C2 ways.

Example (continued) By the Fundamental Counting Principle, the total number of committees is

Example (continued) b. What is the probability that a committee of 5, chosen at random from the class, consists of 3 girls and 2 boys? The total number of committees of 5 is 22C5 = 26,334. From part (a), we know that there are 9900 ways to choose 3 girls and 2 boys.

Example (continued) c. What is the probability that a committee of 5, chosen at random from the class, have no boys? The total number of ways to choose 5 girls and 0 boys from the 12 girls in the class is

Example (continued) d. What is the probability that a committee of 5, chosen at random from the class, consists of only girls? The total number of committees of 5 is 22C5 = 26,334. From part (c), we know that there are 792 ways to choose 5 girls and 0 boys.