728 CHAPTER 11 Probability BOB are possible? If the two Bs were distinguishable (one red and the other blue), there would be six permutations. B B BB BB B B BB BB O O O O O O However, if the Bs are not distinguishable (replacing all red and blue Bs with black print), we see that there are only three permutations. BOB BBO OBB The number of permutations of the letters in BOB can be computed as 3! 2! 3 2 1 2 1 3 = ⋅ ⋅ ⋅ = where 3! represents the number of permutations of three letters, assuming that none are duplicates, and 2! represents the number of ways the two items that are duplicates can be arranged (BB or BB). In general, we have the following rule. Permutations of Duplicate Items The number of distinct permutations of n objects where n1 of the objects are identical, n2 of the objects are identical, n …, r of the objects are identical is determined by using n n n n ! ! ! !r 1 2 Example 8 Duplicate Letters In how many different ways can the letters of the word “CINCINNATI” be arranged? Solution Of the 10 letters, three are I’s, three are N’s, and two are C’s. The number of possible arrangements is 10! 3!3!2! 109876 54 321 3 2 1 3 2 1 2 1 1094754 50,400 4 = ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ ⋅ = There are 50,400 different possible arrangements of the letters in the word “CINCINNATI.” 7 Now try Exercise 51 Exercises Warm Up Exercises In Exercises 1–8, fill in the blank with an appropriate word, phrase, or symbol(s). 1. Any ordered arrangement of a given set of objects is called a(n) ____ _______ . Permutation 2. To determine the number of distinct outcomes when two or more experiments are performed, the fundamental ___________ principle can be used. Counting 3. The symbol for n factorial is ___________. n! 4. The formula for the number of permutations of n distinct items is n! = ___________. n n n ( 1)( 2) (3)(2)(1) − − 5. The formula for the number of permutations when r objects are selected from n objects is Pn r = _______. 6. The number of permutations of n objects, where n1 of the items are identical, n2 of the items are identical, n, r … are identical, is determined by ___________. n n n n ! ! ! !r 1 2 SECTION 11.7 Instructor Resources for Section 11.7 in MyLab Math • Objective-Level Videos 11.7 • PowerPoint Lecture Slides 11.7 • MyLab Exercises and Assignments 11.7 n n r ! ( )! − Answers: 1. a) 60 b) OZONE 2. a) 120 b) BREEZE 3. a) 1260 b) COLLEGE RECREATIONAL MATH Jumble Read the material on Permutations of Duplicate Items before working this JUMBLE. In the Recreational Math box on page 693 we introduced the JUMBLE puzzles. Sometimes the puzzles contain one or more letters multiple times. For each JUMBLE below, use the knowledge you have learned in this section to determine (a) the number of possible arrangements of the letters given. (b) the word that results when the letters are placed in their proper order (note that only one word is possible). 1. NOOZE 2. ZBEERE 3. LOCGEEL The answers are listed upside down below. See Exercises 61 and 62 for more examples.
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