I was just attempting to do the following question:
How many ways can n married couples sit at a round table in such a way that there is one man between every two women and no man is seated next to his wife?
I immediately thought about the well known cyclic permutations formula, which says that if there are n distinct numbers in a circle, there are $(n-1)!$ distinct cyclic permutations of them. I thought that maybe we could use this for each of the two women and then for the man. Unfortunately, I tried it and I couldn't make it to work out. I then thought of the generalized inclusion exclusion principle, but I couldn't think of anything conclusive there either. I have been looking at it for a while now, could you please explain to me how it can be solved using the inclusion-exclusion principle? Moreover, if there exists a simple solution without it, could you please show me how it is solved like that as well? This problem is similar to the problemes des menages by E. Loucas. I have looked for it on the internet, however, I haven't managed to understand any of the solutions