AMC12 2015 B

Answer (D): To make the analysis easier, suppose first that everyone gets up and moves to the chair directly across the table. The reseating rule now is that each person must sit in the same chair or in an adjacent chair. There must be either 0, 2, 4, or 6 people who choose the same chair; otherwise there would be an odd-sized gap, which would not permit all the people in that gap to sit in an adjacent chair. If no people choose the same chair, then either everyone moves left, which can be done in 1 way, or everyone moves right, which can be done in 1 way, or people swap with a neighbor, which can be done in 2 ways, for a total of 4 possibilities. If two people choose the same chair, then they must be either directly opposite each other or next to each other; there are $3+6=9$ such pairs. The remaining four people must swap in pairs, and that can be done in just 1 way in each case. If four people choose the same chair, there are 6 ways to choose those people and the other two people swap. Finally, there is 1 way for everyone to choose the same chair. Therefore there are $4+9+6+1=20$ ways in which the reseating can be done.