1090 CHAPTER 11 Further Topics in Algebra N L M N L M N L M N M L M L N Figure 14 This screen shows how the TI-84 Plus calculates P13, 32 and P13, 22. See Example 4. EXAMPLE 4 Using the Permutations Formula Evaluate. (a) The number of permutations of the letters L, M, and N (b) The number of permutations of 2 of the letters L, M, and N SOLUTION (a) Use the formula for P1n, r2, with n = 3 and r = 3. P13, 32 = 3! 13 - 32! = 3! 0! = 3 # 2 # 1 1 = 6 As shown in the tree diagram in Figure 14, the 6 permutations of the letters are as follows. LMN, LNM, MLN, MNL, NLM, NML (b) Evaluate P13, 22. P13, 22 = 3! 13 - 22! = 3! 1! = 3 # 2 # 1 1 = 6 This result is the same as the answer in part (a). After the first two letter choices are made, the third is already determined because only one letter is left. S Now Try Exercise 51. The result in Example 4(a) can be generalized for all n. P1n, n2 =n! EXAMPLE 5 Using the Permutations Formula Suppose 8 people enter an event in a swim meet. In how many ways could the gold, silver, and bronze medals be awarded? SOLUTION Using the fundamental principle of counting, there are 3 choices to be made, giving 8 # 7 # 6 = 336 ways. We can also use the formula for P1n, r2 to obtain the same result. P18, 32 = 8! 18 - 32! Permutations formula with n = 8 and r = 3 = 8! 5! Subtract in the denominator. = 8 # 7 # 6 # 5 # 4 # 3 # 2 # 1 5 # 4 # 3 # 2 # 1 Definition of n! = 8 # 7 # 6 Divide out the common factors. = 336 ways Multiply. S Now Try Exercise 49.
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