2008 AMC 10B Problems/Problem 21
Contents
[hide]Problem
Ten chairs are evenly spaced around a round table and numbered clockwise from through . Five married couples are to sit in the chairs with men and women alternating, and no one is to sit either next to or across from his/her spouse. How many seating arrangements are possible?
Solution
For the first man, there are possible seats. For each subsequent man, there are , , , or possible seats. After the men are seated, there are only two possible arrangements for the five women. The answer is .
Solution 2
Label the seats ABCDEFGHIJ, where A is the top seat. The first man has possible seats. WLOG, assume he is in seat A in the diagram. Then, his wife can be in one of two seats, namely D or H. WLOG, assume she is in seat D. Now, in each structurally distinct solution we find, we know that there are ways to arrange the 4 other couples. Let there be x structurally distinct solutions under these conditions. We know the answer must be possible seating arrangements, where x is a nonnegative integer. There is only one answer that is a multiple of . So, our answer is .
See also
2008 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 20 |
Followed by Problem 22 | |
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All AMC 10 Problems and Solutions |
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