In a convex $n$-sided polygon, all the diagonals are drawn and no three of them pass through a point. Find a formula for the number of regions formed inside the polygon.
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

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