Problem 690

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 $\left(\begin{array}{l}n \\ 2\end{array}\right)-n$, since each diagonal corresponds to a pair of non-adjacent vertices. The number of intersections of two diagonals is $\left(\begin{array}{l}n \\ 4\end{array}\right)$, since each set of four vertices determines a unique pair of intersecting diagonals. Thus, the total number of regions formed is

$$1+\left(\begin{array}{l} n \\ 2 \end{array}\right)-n+\left(\begin{array}{l} n \\ 4 \end{array}\right)$$