$4$ married couples and $2$ single men sit at a circular table. In how many ways can they sit so that a man can't sit next to a woman who is not his wife?
I have tried but I am not sure to the following answer :
First, the husbands sit in the circular table of which the possible ways is $3!$.
The wives should sit next to her husband which is only $1$ way.
The two men can only be put between two husbands in only $2$ ways and they then can be permuted in $2$ ways. So the number of ways is $4$.
By applying the multiplication principle, the total number of ways for this possibility is $3! \times 4$ = $24$.
The second possibility is similar which is making the wives sit first and put each husband next to his own wife and then put the single men. The total number of ways is also $24$.
So the answer is $24 + 24 = 48$.