Example 4. Connecting every pair of $n$ points on a circle, with no three lines intersecting at the same point inside the circle, how many regions do they divide the circle into?


Consider the intersection points of these lines inside the circle. Each such intersection point corresponds to four points on the circumference, and vice versa, so the number of intersection points is $C_{n}^{4}$. It is also clear that the number of lines is $C_{n}^{2}$, so the number of regions they divide the circle into is $S=C_{n}^{4}+C_{n}^{2}+1$.