11.7 The Fundamental Counting Principle and Permutations 723 Permutations Now we introduce the definition of a permutation. Example 3 Garage Door Codes Many garage doors come equipped with a keypad that allows residents to open the garage door by entering a four-digit code. Repetition of digits in the code is allowed. a) How many different four-digit codes are possible? b) If one of these codes is randomly selected, what is the probability that the code has no repeated digits? Solution a) Since repetition of digits is allowed, there are 10 choices for each of the four digits. Therefore, using the fundamental counting principle, there are 10 10 10 10 10,000 ⋅ ⋅ ⋅ = possible four-digit codes. b) Of these 10,000 possible codes, we need to determine how many codes have no repeated digits. There are 10 choices for the first digit, and since the digits do not repeat, there are 9 choices for the second digit, 8 choices for the third digit, and 7 choices for the fourth digit. Using the fundamental counting principle, there are 10 9 8 7 5040 ⋅ ⋅ ⋅ = codes with no repeated digits. We calculate the probability as follows. P(no repeated digits) number of codes with no repeated digits number of possible codes 5040 10,000 63 125 0.504 = = = = Thus, the probability that a randomly selected four-digit garage code has no repeated digits is 0.504. 7 Now try Exercise 35 Definition: Permutation A permutation is any ordered arrangement of a given set of objects. “Superman, Batman, Wonder Woman” and “Wonder Woman, Superman, Batman” represent two different ordered arrangements or two different permutations of the same three characters. In Example 2(a), there are 120 different ordered arrangements, or permutations, of the five different albums. In Example 2(b), there are 24 different ordered arrangements, or permutations possible, if the Styx album must be displayed in the middle. When determining the number of permutations possible, we assume that repetition of an item is not permitted. To help you understand and visualize permutations, we illustrate the various permutations possible when a triangle, rectangle, and circle are to be placed in a line; see Fig. 11.19. , , , , , Figure 11.19 For this set of three shapes, six different arrangements, or six permutations, are possible. We can obtain the number of permutations by using the fundamental counting TCD/Prod.DB/Alamy Stock Photo m Batman, Wonder Woman, and Superman
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