Solution:

To start with, there is one region inside the $n$-gon. A new region is created each time a new diagonal is drawn or two diagonals intersect. The number of diagonals is $\binom{n}{2} - n$, since each diagonal corresponds to a pair of non-adjacent vertices. The number of intersections of two diagonals is $\binom{n}{4}$, since each set of four vertices determines a unique pair of intersecting diagonals. Thus, the total number of regions formed is

$$1 + \binom{n}{2} - n + \binom{n}{4}$$