355. Diagonals of a Polygon. Find the greatest possible number of intersections of the diagonals of a plane convex $n$-gon.
355. Any four vertices completely determine some quadrilateral. The only point of intersection of the diagonals of such a quadrilateral lies inside the original polygon due to convexity. This point belongs to exactly two diagonals of our $n$-gon. Therefore, the number of diagonal intersections in the original $n$-gon will be maximal when all such points are distinct. In this case, their number will coincide with the number of quadrilaterals that can be formed from the vertices of the original polygon, which is

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

[R. Hude, M. M., 45, 284 (May 1972).]