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2001 AMC 12 Problem 16

Problem 16 of 25IntermediateCombinatorics

A spider has one sock and one shoe for each of its eight legs. In how many different orders can the spider put on its socks and shoes, assuming that, on each leg, the sock must be put on before the shoe?

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Solution

Think of the 1616 items (88 socks and 88 shoes) arranged in some order: there are 16!16! arrangements. For each leg, the sock comes before the shoe in exactly half of all arrangements. Imposing this on all eight legs independently divides by 28,2^8, giving 16!28. \dfrac{16!}{2^8}. Thus, the correct answer is D.
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Tagged: arrangements with restrictions · permutations

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