Question 83
An apartment building consists of 20 rooms numbered $1, 2, \ldots, 20$ arranged clockwise in a circle. To move from one room to another, one can either walk to the next room clockwise (i.e. from room $i$ to room $(i+1) \bmod 20$) or walk across the center to the opposite room (i.e. from room $i$ to room $(i+10) \bmod 20$). Find the number of ways to move from room 10 to room 20 without visiting the same room twice.