A Problem Solving Approach to Mathematics for Elementary

Section 9 -4 Counting Techniques in Probability • The Fundamental Counting Principle • Permutations of like objects. • Combinations. • Use of counting techniques in probability problems. Copyright © 2020, 2016, 2012 Pearson Education, Inc. All Rights Reserved Slide - 2

Permutations of Unlike Objects Permutation 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 ways. Copyright © 2020, 2016, 2012 Pearson Education, Inc. All Rights Reserved Slide - 3

Factorial n factorial Copyright © 2020, 2016, 2012 Pearson Education, Inc. All Rights Reserved

Permutations of Objects in a Set The number of permutations of r objects chosen is denoted by from a set of n objects, where and is given by Copyright © 2020, 2016, 2012 Pearson Education, Inc. All Rights Reserved Slide - 5

Example 17 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. Copyright © 2020, 2016, 2012 Pearson Education, Inc. All Rights Reserved Slide - 6

Permutations Involving Like Objects If there are n objects, of which are alike, and so on through then the number of different arrangements of all n objects, where alike objects are indistinguishable, is equal to Copyright © 2020, 2016, 2012 Pearson Education, Inc. All Rights Reserved Slide - 7

Example 18 Find the number of rearrangements of the letters in each of the

Combinations (1 of 2) Combination An arrangement of objects in which the order makes

Combinations (2 of 2) The number of combinations of r objects chosen from a set of n objects, where 0 ≤ r ≤ n, is denoted and is given by: Copyright © 2020, 2016, 2012 Pearson Education, Inc. All Rights Reserved Slide - 10

Example 19 (1 of 2) 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 ways to choose the free books. The three circled numbers can be arranged in ways. Copyright © 2020, 2016, 2012 Pearson Education, Inc. All Rights Reserved Slide - 11

Example 19 (2 of 2) Copyright © 2020, 2016, 2012 Pearson Education, Inc. All

Example 20 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. Copyright © 2020, 2016, 2012 Pearson Education, Inc. All Rights Reserved Slide - 13

Example 21 (1 of 5) 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 ways. The boys can be chosen in ways. Copyright © 2020, 2016, 2012 Pearson Education, Inc. All Rights Reserved Slide - 14

Example 21 (2 of 5) By the Fundamental Counting Principle, the total number of

Example 21 (3 of 5) 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 From part (a), we know that there are 9900 ways to choose 3 girls and 2 boys. Copyright © 2020, 2016, 2012 Pearson Education, Inc. All Rights Reserved Slide - 16

Example 21 (4 of 5) 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 Copyright © 2020, 2016, 2012 Pearson Education, Inc. All Rights Reserved Slide - 17

Example 21 (5 of 5) 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 From part (c), we know that there are 792 ways to choose 5 girls and 0 boys. Copyright © 2020, 2016, 2012 Pearson Education, Inc. All Rights Reserved Slide - 18