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2002 AMC 10B Problem 9

Problem 9 of 25EasierCombinatoricsProblem-Solving Techniques

Using the letters A,A, M,M, O,O, S,S, and U,U, we can form 120120 five-letter “words.” If these “words” are arranged in alphabetical order, then the “word” USAMOUSAMO occupies which position?

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Solution

The alphabetical order of the letters is A,M,O,S,U.A, M, O, S, U. Words beginning with A,A, M,M, O,O, or SS fill positions 11 through 9696 (four choices of first letter, 2424 each). Words beginning with UU occupy positions 9797–120.120. Listing them alphabetically, USAMOUSAMO is the 1919th such word, so it occupies position 96+19=115.96 + 19 = 115. Thus, the correct answer is D.
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Tagged: permutations · systematic listing

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