# How do I tackle this combinatorics problem about married couples around a table?

Here is the question, from Bogart's :

A group of n married couples comes to a group discussion session where they all sit around a round table. In how many ways can they sit so that no person is next to his or her spouse? (Note that two people of the same sex can sit next to each other.

I appreciate any tips or advice.

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This is a variant of the Menage Problem and a solution of that can be found in one of Bogart's articles(no surprise there) itself: http://www.math.dartmouth.edu/~doyle/docs/menage/menage/menage.html

(See the solution to the Relaxed Menage Problem).

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Thank You So Much - this clears it up a lot! –  Adel Mar 6 '12 at 22:54
@Adel: You are welcome :-) –  Aryabhata Mar 6 '12 at 23:03

I don't understand why " we begin with the set S of all (2n)! ways of seating the 2n individuals around the table" (from the article)...isn't it (2n-1)! options?

thank you.

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I think in that part of the solution, you don't consider circular rotations as equivalent: label the seats $1$, $2$, $\dots$, $2n$, then you're looking at $(2n)!$ ways of assigning the $2n$ individuals to the $2n$ seats. –  ShreevatsaR Mar 27 '14 at 15:37
Actually you don't consider rotations as equivalent even in the final problem considered there. –  ShreevatsaR Mar 27 '14 at 15:43