Suppose that $\displaystyle n $ points lie on a circle are all joined in pairs. The points are positioned so that no three joining lines are concurrent in the interior of the circle. Let $\displaystyle a_n $ be the number of regions into which the interior of the circle is divided. Draw diagrams to find $\displaystyle a_n $ is given by the following formula $\displaystyle a_n = n + \binom{n-1}{2} + \binom{n-1}{3} + \binom{n-1}{4} = 1+ \frac{n(n-1)(n^{2}-5n+18)}{24} $.

I am pretty sure that I can do the induction part. I tried plugging in $\displaystyle n = 2,3 $ and the number of regions were 2 and 4. The diagrams also had 2 and 4 regions respectively. But when I plugged in $\displaystyle n = 4 $ $\displaystyle a_n = 8 $. However, when I drew it I got $\displaystyle a_n = 9 $.

Here are the circles:

Am I drawing the points wrong in the third circle?