Question 9
Let $S$ be a finite set of points in the plane such that no three of them are on a line. For each convex polygon $P$ whose vertices are in $S$, let $a(P)$ be the number of vertices of $P$, and let $b(P)$ be the number of points of $S$ which are outside $P$. Prove that for every real number $x$
$$
\sum_{P} x^{a(P)}(1-x)^{b(P)}=1,
$$
where the sum is taken over all convex polygons with vertices in $S$.
NB. A line segment, a point and the empty set are considered as convex polygons of 2, 1 and 0 vertices, respectively.
$$
\sum_{P} x^{a(P)}(1-x)^{b(P)}=1,
$$
where the sum is taken over all convex polygons with vertices in $S$.
NB. A line segment, a point and the empty set are considered as convex polygons of 2, 1 and 0 vertices, respectively.