11.7 The Fundamental Counting Principle and Permutations 725 In Example 5, we determined the number of different ways in which we could select and arrange three of the five items. We can indicate that result by using the notation P . 5 3 The notation P5 3 is read “the number of permutations of five items taken three at a time.” The notation Pn r is read “the number of permutations of n items taken r at a time.” We use the fundamental counting principle below to evaluate P P , , 8 4 9 3 and P . 10 5 Note the relationship between the number preceding the P, the number following the P, and the last number in the product. P P P 8 7 6 5 9 8 7 10 9 8 7 6 8 4 9 3 10 5 = ⋅ ⋅ ⋅ = ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ To evaluate P , n r we begin with n and form a product of r consecutive decreasing factors. For example, to evaluate P , 10 5 we start with 10 and form a product of five consecutive decreasing factors (see the preceding illustration). In general, the number of permutations of n items taken r at a time, P , n r may be determined by the formula P n n n n r ( 1)( 2) ( 1) n r = − − − + Therefore, when evaluating P , 20 15 we would determine the product of consecutive decreasing integers from 20 to (20 15 1) − + or 6, which is written as 20 19 18 17 6. ⋅ ⋅ ⋅ ⋅ ⋅ Now let’s develop an alternative formula that we can use to determine the number of permutations possible when r objects are selected from n objects: P n n n n r ( 1)( 2) ( 1) n r = − − − + Now multiply the expression on the right side of the equals sign by n r n r ( )! ( )! , − − which is equivalent to multiplying the expression by 1. P n n n n r n r n r ( 1)( 2) ( 1) ( )! ( )! n r = − − − + × − − For example, P 10 9 6 5! 5! 10 5 = ⋅ ⋅ ⋅ × or P 10 9 6 5! 5! 10 5 = ⋅ ⋅ ⋅ × Since n r ( )! − means n r n r ( )( 1) (3)(2)(1), − − − the expression for Pn r can be rewritten as = − − − + − − − − − P n n n n r n r n r n r ( 1) ( 2) ( 1) ( ) ( 1) (3) (2) (1) ( )! n r n r ( )! Since the numerator of this expression is n!, we can write P n n r ! ( )! n r = − For example, P 10! (10 5)! 10 5 = − Now we give the permutation formula. One more than 8 4 − One more than 9 3 − One more than 10 5 − One more than n r − RECREATIONAL MATH Stock Ticker Symbols Stock ticker symbols on the New York Stock Exchange (NYSE) typically consist of one, two, or three letters. For example, T represents AT&T, GE represents General Electric, and IBM represents International Business Machines. Stocks on the National Association of Security Dealers Automated Quotation System (NASDAQ) typically contain four letters. For example, AAPL represents Apple and GOOG represents Alphabet, the parent company of Google. Are there more possible NYSE ticker symbols or more NASDAQ ticker symbols? The answers are listed upside down below. Answer: For the NYSE there are 26 (26 26) + × + (26 26 26) 18,278 × × = different possible ticker symbols. For the NASDAQ there are 26 26 26 26 456,976 × × × = different possible ticker symbols.
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