Question 200
For every positive integer $n$, find (with proof) the number of ways of sitting $2n$ people $A_1, B_1, A_2, B_2, \dots, A_n, B_n$ around a round table so that $A_i$ and $B_i$ are not next to each other for $i = 1, 2, \dots, n$. Two sitting arrangements are considered the same if one can be obtained from another by a rotation. (You may express your answer in whatever ways.)