I've been trying to solve the following home task:
Choose $n$ points ($n\ge 2$) on the circle's circumference and connect them all with each other using chords. In result, the circle is divided into $R$ number of regions. Let $P$ be the number of intersection points and $S$ the number of (chord) segments.
Prove that $P-S+R=1$.
A hint is provided that it's not advised nor needed to try and find the exact counts for $P$, $S$ or $R$.
My first insight is that we can always create a regular polygon with $n$ sides if we put the points on circle equally spaced. So for example a pentagon has $P=5$, $S=20$ and $R=16$.
But here I'm stuck badly. Maybe we could apply Euler's polyhedron formula but I don't see how since it's for polyhedrons. The hint makes me think about an inductive approach. But if we start with $n$ points and add one more (that is, go from $n$-gon to ($n+1$)-gon), I don't see how we can make use of the induction assumption to proof the $(n+1)$ case.
Any insight is welcome!