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Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 9.5- 2 NCTM Standard: Data Analysis and Probability  K–2: Children should discuss events related to their experience as likely or unlikely. (p. 400)  3–5: Children should be able to “describe events as likely or unlikely and discuss the degree of likelihood using words such as certain, equally likely, and impossible.” They should be able to “predict the probability of outcomes of simple experiments and test the predictions.” They should “understand that the measure of the likelihood of an event can be represented by a number from 0 to 1.” (p. 400)

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 9.5- 3 NCTM Standard: Data Analysis and Probability 6–8: Children should “understand and use appropriate terminology to describe complementary and mutually exclusive events.” They should be able “to make and test conjectures about the results of experiments and simulations.” They should be able to “compute probabilities of compound events using methods such as organized lists, tree diagrams, and area models.” (p. 401)

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 9.5- 5 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 m · n ways.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 9.5- 7 Permutations of Objects in a Set In a set of n elements, the number of ways to choose elements from the set in order, the permutations of n objects taken r at a time, is given by

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 9.5- 8 a.A baseball team has nine players. Find the number of ways the manager can arrange the batting order. Example 9-17 b.Find the number of ways of choosing three initials from the alphabet if none of the letters can be repeated.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 9.5- 9 Permutations Involving Like Objects If a set contains n elements, of which r 1 are of one kind, r 2 are of another kind, and so on through r k, then the number of different arrangements of all n elements is equal to

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 9.5- 10 Find the number of rearrangements of the letters in each of the following words: Example 9-18 a.bubble b.statistics

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 9.5- 12 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.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 9.5- 13 Example 9-19 The Library of Science Book Club offers 3 free books from a list of 42. If you circle 3 choices from a list of 42 numbers representing the book on a postcard, how many possible choices 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.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 9.5- 15 Example 9-20 At the beginning of the second quarter 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 © 2010 Pearson Education, Inc. All rights reserved. Slide 9.5- 16 Example 9-21 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 12 C 3 ways. The boys can be chosen in 10 C 2 ways.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 9.5- 18 Example 9-21 (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 22 C 5 = 26,334. From part (a), we know that there are 9900 ways to choose 3 girls and 2 boys.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 9.5- 19 Example 9-21 (continued) c.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 ways to choose 5 girls and 0 boys from the 12 girls in the class is

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 9.5- 20 Example 9-21 (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 22 C 5 = 26,334. From part (c), we know that there are 792 ways to choose 5 girls and 0 boys.