1093 11.6 Basics of Counting Theory (b) Order is not important, so we use combinations to select 2 of the 11 women and 2 of the 19 men. C111, 22 # C119, 22 = 11! 2!9! # 19! 2!17! Use combinations and the fundamental principle of counting. = 55 # 171 Evaluate. = 9405 Multiply. In this case, the project group can be selected in 9405 ways. S Now Try Exercise 61. Characteristics of Permutations and Combinations Permutations Combinations These are selections of r items from n items. Repetitions are not allowed. Order is important. Order is not important. These are arrangements of r items from a set of n items. These are subsets of r items from a set of n items. P1n, r2 = n! 1 n −r2! C1n, r2 = a n rb = n! r!1n −r2! Clue words: arrangement, schedule, order Clue words: group, committee, sample, selection These are combinations. The order of the cards in the hands is not important. Characteristics That Distinguish Permutations from Combinations Consider the following table. EXAMPLE 9 Distinguishing Permutations and Combinations Determine whether permutations or combinations should be used to solve each problem. (a) How many 4-digit codes are possible if no digits are repeated? (b) A sample of 4 light bulbs is randomly selected from a batch of 15 bulbs to be packaged and sold. How many different samples are possible? (c) In a basketball tournament with 8 teams, how many games must be played so that each team plays every other team exactly once? (d) In how many ways can 4 stockbrokers be assigned to 6 offices so that each broker has a private office? SOLUTION (a) Changing the order of the 4 digits results in a different code, so permutations should be used. (b) The order in which the 4 light bulbs are selected is not important. The sample is unchanged if the items are rearranged, so combinations should be used. (c) Selection of 2 teams for a game creates an unordered subset of 2 from the set of 8 teams. Use combinations. (d) The office assignments are an ordered selection of 4 offices from the 6 offices. Exchanging the offices of any 2 brokers within a selection of the 4 offices gives a different assignment, so permutations should be used. S Now Try Exercise 35.
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