It is known that in a convex $n$-gon ($n>3$) no three diagonals pass through the same point.

Find the number of points (distinct from the vertices) of intersection of pairs of diagonals.
Each pair of intersecting diagonals can be associated with a quadruple of vertices of the $n$-gon, which are the endpoints of these diagonals.

## Solution

The problem can be restated as follows: we need to find the number of pairs of intersecting diagonals of a convex $n$-gon. Each pair of intersecting diagonals can be associated with a quadruple of vertices of the $n$-gon, which are the endpoints of these diagonals. Conversely, for any quadruple of vertices, there is exactly one pair of intersecting diagonals whose endpoints are these four vertices. Therefore, the number of pairs of intersecting diagonals is equal to the number of ways to choose four vertices from $n$.

## Answer

$C_{n}^{4}=\frac{n(n-1)(n-2)(n-3)}{24}$