Ways to arrange $n\geq2$ people around a circular table, given two permanent seats. How many ways to arrange $n\geq2$ people around a circular table, given two specific people who cannot stand next to each other?
I've observed that when $n=2$ and $n=3$ there exists no way to arrange them so that the two specific people aren't standing next to each other. I think for when $n=4$ there are $3!$ ways to arrange them, but I've had trouble coming up with a formula that can describe all values of $n$.
Perhaps I can attack the problem by first arranging the other people in the circle, and then placing either one of the two specific people in between the gaps?

Well there are $n$ seats. Assuming the seats are not numbered, so the first mandatory person can be placed anywhere.  
Then, place the second mandatory person, there are $n-3$ choices (why?)
Place the rest of $n-2$ people into $n-2$ seats, so $(n-2)!$ ways
Total: $(n-3)(n-2)!$ ways
