Choose $n$ points on a circle so that no three of the $\binom{n}{2}$ chords have a common point inside the circle. Let $a_{n}$ be the number of regions formed inside the circle by drawing the cords.
Obtain the recurrence relatin $a_{n}=a_{n-1}+f(n)$ for $n\geq 1$ where $f(n)=n-1+\sum_{i=1}^{n-1}(i-1)(n-1-i)$.
It is clear where the $a_{n-1}$ comes from and the $n-1$ in $f(n)$ but what does this product mean? We have $(i-1)(n-1-i)$ extra regions for each $i$? And what would the $i$ be indexing? The points on the circle? Any hints on this problem would be nice. I've been stumped for a few days now.