Suppose that 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 be the number of regions into which the interior of the circle is divided. Draw diagrams to find is given by the following formula .

I am pretty sure that I can do the induction part. I tried plugging in and the number of regions were 2 and 4. The diagrams also had 2 and 4 regions respectively. But when I plugged in . However, when I drew it I got .

Here are the circles:

Am I drawing the points wrong in the third circle?