I am going through a solution of the following problem. "How many ways there are there to seat $n$ couples around a circular table with $2n$ chairs such that no couple sits next to each other, i.e., no husband and wife sit adjacent to each other."
- The solution starts with the following : The number of possible arrangements where the men and women alternate is $2(n!)^2$, because, we can seat the women in $2n!$ ways, and for each such choice there are $n!$ ways to seat the men.
According to me this is true for linear permutation, not in case of circular permutation. in case of circular permutation the answer should be $n[(n-1)!]^2 $. Am I correct ?