Let there be $n$ pairs of husband-wife, and a round table with $2n$ chairs. Suppose that $n$ wives are already sat down, and between any two neighboring wives there is exactly one free chair (there is an alternation of empty chairs and occupied wives, $\dots,\text{chair}_n,\text{wife}_1,\text{chair}_1,\text{wife}_2, \text{chair}_2,\dots$). Find the number of $n$ husband's placing on the remaining chairs, where only $r$ husbands, $ 0 \leq r \leq n $, are directly near with their wives?
I think, about Inclusion–exclusion principle in this issue; we can called $\alpha_i$ - "$i$-th man sitting with his wife". Then look at $ \alpha_{i_1}\alpha_{i_2}\cdots\alpha_{i_r}$ - "$\geq r$ husbands sitting near their wives": $$ n! - |\bar\alpha_{i_1}\bar\alpha_{i_2}\cdots\bar\alpha_{i_r}| = |\alpha_{i_1}\cup\alpha_{i_2}\cup\dots\cup\alpha_{i_r}| = |\alpha_{i_1}| + |\alpha_{i_2}| +\dots + |\alpha_{i_r}| - |\alpha_{i_1}\alpha_{i_2}| - |\alpha_{i_1}\alpha_{i_3}| - \dots - |\alpha_{i_r}\alpha_{i_{r-1}}| + \dots + (-1)^{r}|\alpha_{i_1}\alpha_{i_2}\cdots\alpha_{i_r}|$$
If we know about $A_{\geq r}$, we can find $A_{=r} = A_{\geq r} - A_{\geq r+1}$. I stuck with calculation of it; or may be my $\alpha_i$ are not adequate for this problem.