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?

We can fix one of the 2 pearson. Then we have $n-3$ positions for the second pearson (since one is already ocupied and it can not be seated near to it), and then we can arrange the other on $(n-2)!$ ways, so the answer is $(n-3)\cdot (n-2)!$
