Number of Ways of Seating People at Circular Table

Theorem
Let there be $n$ people to be seated at a round table.

Let $N$ be the number of different ways to seat those $n$ people.

Note that if person $A$ is seated between person $B$ and person $C$, then having $B$ on the left and $C$ on the right is considered different from having $B$ on the right and $C$ on the left.

Then:
 * $N = \paren {n - 1}!$

Proof
From Number of Permutations of All Elements, the number of different ways of arranging $n$ people in line is $n!$.

However, a round table has no beginning and end.

Hence the vital factor is the arrangement relative to an arbitrary given person.

So we fix one person, and arrange the other $n - 1$ people relative to that person.

From Number of Permutations of All Elements, there are $\paren {n - 1}!$ ways to arrange those people.