Number of Different n-gons that can be Inscribed in Circle

Theorem
Let $C$ be a circle on whose circumference $n$ points are placed which divide $C$ into $n$ equal arcs.

The number of different $n$-gons (either stellated or otherwise) that can be described on $C$ whose vertices are those $n$ points is:


 * $S_n = \dfrac {\paren {n - 1}!} 2$

Proof
An $n$-gon (either stellated or otherwise) is constructed by connecting all $n$ points in some order.

From Number of Permutations, there are ${}^n P_n = n!$ ways of ordering $n$ points.

However, for each $n$-gon formed in this way, we have:
 * $n$ ways of choosing the first vertex
 * $2$ different sides from that vertex to use as the first one (starting off either clockwise or anticlockwise).

and all of these end up describing the same $n$-gon.

Thus there are $\dfrac n 2 n! = \dfrac {\paren {n - 1}!} 2$ different $n$-gons.