Subject: Circular Permutations

Name: J.Stuart

Who is asking: Student
Level: All

I am able to understand the simple theory behind circular permutations (that is, there are (n-1)! ways to arrange n people in a circle. One question, however, is giving me some difficulty. It reads: In how many ways can four married couples be arranged around a circular table if each man must sit beside his wife? Any suggestions you have would be greatly appreaciated.
Thanks, JS


Why not think about seating the women first, say in a cyclic order 1,2,3,4. I no woman sits adjacent to another then this defines 4 spaces between consecutive women for the men to be seated. Pick the first man, spouse of #1; in how many ways can he be seated? What about the second man, spouse of #2??

Two other situations are possible. The first is that exactly two women sit adjacent to each other and the second that there are two pairs of women who sit adjacent to each other. In both cases you can analyze how the men can be arranged.

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