Exercise C.1.3

In how many ways can $n$ professors sit around a circular conference table? Consider two seatings to be the same if one can be rotated to form the other.

If we pick a chair as the first one and a direction, in which the chairs are increasing, there are $n!$ ways to seat the professors. However, an arrangement $\{1, 2, \ldots, n\}$ is identical to $\{2, 3, \ldots, n, 1\}$ and there are $n$ of those identical arrangements (one starting on each professor). Since identical arrangements are in groups of size $n$, the total number of seatings is $n!/n = (n-1)!$.