170 CHAPTER 3 Probability You may want to order a group of n objects in which some of the objects are the same. For instance, consider the group of letters AAAABBC. This group has four A’s, two B’s, and one C. How many ways can you order such a group? Using the formula fornPr, you might conclude that there are 7P7 = 7! = 5040 possible orders. However, because some of the objects are the same, not all of these permutations are distinguishable. How many distinguishable permutations are possible? The answer can be found using the formula for the number of distinguishable permutations. The number of distinguishable permutations of n objects, where n1 are of one type, n2 are of another type, and so on, is n! n1! # n2! # n3! gnk! where n1 + n2 + n3 + g+ nk = n. Distinguishable Permutations Using the formula for distinguishable permutations, you can determine that the number of distinguishable permutations of the letters AAAABBC is 7! 4! # 2! # 1! = 7# 6# 5 2 = 105 distinguishable permutations. Finding the Number of Distinguishable Permutations A building contractor is planning to develop a subdivision. The subdivision is to consist of 6 one-story houses, 4 two-story houses, and 2 split-level houses. In how many distinguishable ways can the houses be arranged? SOLUTION There are to be 12 houses in the subdivision, 6 of which are of one type (one-story), 4 of another type (two-story), and 2 of a third type (split-level). So, there are 12! 6! # 4! # 2! = 12# 11# 10# 9# 8# 7# 6! 6! # 4! # 2! = 13,860 distinguishable ways. You can check your answer using technology, as shown at the left on a TI-84 Plus. Interpretation There are 13,860 distinguishable ways to arrange the houses in the subdivision. TRY IT YOURSELF 4 The contractor wants to plant six oak trees, nine maple trees, and five poplar trees along the subdivision street. The trees are to be spaced evenly. In how many distinguishable ways can they be planted? Answer: Page A38 EXAMPLE 4 TI-84 PLUS 12!/(6!4!2!) 13860
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