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2008 AMC 10B Problem 21

Problem 21 of 25HarderCombinatoricsProblem-Solving Techniques

Ten chairs are evenly spaced around a round table and numbered clockwise from 11 through 10.10. Five married couples are to sit in the chairs with men and women alternating, and no one is to sit either next to or directly across from his or her spouse. How many seating arrangements are possible?

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Solution

Seat the women first. The first woman may take any of the 1010 chairs, and since seats alternate, the remaining women fill their four seats in 4!4! ways, giving 10⋅4!=24010\cdot 4!=240 arrangements. Fix a woman in chair 1.1. Her spouse must sit in chair 44 or chair 8;8; each choice then forces the placement of every other man consistently. So each seating of the women yields exactly 22 valid seatings of the men. The total is 2⋅240=480.2\cdot 240=480. Thus, the correct answer is C.
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