1092 CHAPTER 11 Further Topics in Algebra Combinations of n Elements Taken r at aTime If C1n, r2 represents the number of combinations of n elements taken r at a time, with r … n, then the following holds true. C1n, r2 = n! 1 n −r2! r! , or C1n, r2 = n! r!1n −r2! Alternative notations for C1n, r2 are An r B and nCr . EXAMPLE 7 Using the Combinations Formula How many different committees of 3 people can be chosen from a group of 8 people? SOLUTION A committee is an unordered set, so use the combinations formula with n = 8 and r = 3. C18, 32 = 8! 3!18 - 32! Combinations formula with n = 8 and r = 3 = 8! 3! 5! Subtract in the denominator. = 8 # 7 # 6 # 5! 3! 5! Definition of factorial = 8 # 7 # 6 6 3! = 6 = 56 committees Divide out the common factor; Multiply. S Now Try Exercise 53. This screen shows how the TI-84 Plus calculates C18, 32. See Example 7. EXAMPLE 8 Using the Combinations Formula A group of stockbrokers consists of 11 women and 19 men. Four will be selected to work on a special project. (a) In how many different ways can the stockbrokers be selected? (b) In how many ways can the group of 4 be selected if 2 must be women and 2 must be men? SOLUTION (a) Here we wish to know the number of 4-element combinations that can be formed from a set of 11 + 19 = 30 elements. (We want combinations, not permutations, because order within the group does not matter.) C130, 42 = 30! 4! 26! = 27,405 There are 27,405 ways to select the project group.
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