# menage problem- another version

find the number of different ways in which it is possible to seat a set of male-female couples at a round table so that men and women alternate and nobody sits next to his or her partner. Two seatings are considered the same if one is a rotation of the other.

I found the solution of this problem both in wikipedia and Bogart's article, but there they don't consider two seatings the same if one is a rotation of the other.

I've tried to use the principle of inclusion and exclusion:

there are $n!(n-1)!$ ways to arrange n women and n men at a round table, when they alteranate, we define: $A_{i}=$ the i couple sitting together.

then $|A_{i}|=2 * ((n-1)!)^2$. now, $|A_{i}$ ang $A_{j}| = ?$ here it's more compicate, because I think we need to consider two different stuations: the 4 are sitting together/ there are people between couples i and j... thank you!

There are $2n$ possible rotations of each permissible configuration, so just divide the previous solution by $2n$.