Stellated polygons

Statement

On a circle we fix $7$ points, one of which is a starting point. A stellated polygon is built by joining the points in any order (edges may cross), beginning at the fixed point. Since each polygon can be traversed in two directions, the number of distinct stellated polygons is $\dfrac{(p-1)!}{2}$ with $p=7$. How many are there?

Solution

With the first vertex fixed, the other $6$ can be arranged in $6!$ orders; dividing by $2$ (the two traversal directions) gives $N=\frac{(7-1)!}{2}=\frac{6!}{2}=\frac{720}{2}=\boxed{360}.$ (This is the combinatorial idea behind Wilson’s theorem.)