SECTION 13.2 Permutations and Combinations 919 THEOREM Permutations of r Objects Chosen from n Distinct Objects without Repetition The number of arrangements of n objects using ≤ r n of them, in which • the n objects are distinct • repetition of objects is not allowed • order is important is given by the formula ( ) ( ) = − P n r n n r , ! ! (1) Computing Permutations Evaluate: (a) ( ) P 7, 3 (b) ( ) P 6, 1 (c) ( ) P 52, 5 Solution EXAMPLE 4 Parts (a) and (b) are each worked two ways. (a) ( ) = ⋅ ⋅ = P 7,3 765 210 3 factors or ( ) ( ) = − = = ⋅ ⋅ ⋅ = P 7, 3 7! 7 3 ! 7! 4! 7 6 5 4! 4! 210 (b) ( ) = = P 6, 1 6 6 1 factor or ( ) ( ) = − = = ⋅ = P 6, 1 6! 6 1 ! 6! 5! 6 5! 5! 6 (c) Figures 3(a) and 3(b) show the solution using a TI-84 Plus CE graphing calculator and Desmos. So ( ) = P 52, 5 311,875,200 Figure 3(a) ( ) P 52, 5 Figure 3(b) Now Work PROBLEM 7 The Birthday Problem All we know about Shannon, Patrick, and Ryan is that they have different birthdays. If all the possible ways this could occur were listed, how many would there be? Assume that there are 365 days in a year. Solution EXAMPLE 5 This is an example of a permutation in which 3 birthdays are selected from a possible 365 days, and no birthday may repeat itself. The number of ways this can occur is ( ) ( ) = − = ⋅ ⋅ ⋅ = ⋅ ⋅ = P 365,3 365! 365 3 ! 365 364 363 362! 362! 365 364 363 48,228,180 There are 48,228,180 ways in which three people can all have different birthdays. 2 Solve Counting Problems Using Combinations In a permutation, order is important. For example, the arrangements ABC CAB , , BAC, … are considered different arrangements of the letters A, B, and C. In many situations, though, order is unimportant. For example, in the card game of poker, the order in which the cards are received does not matter; it is the combination of the cards that matters. Now Work PROBLEM 47
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