What is the maximum number of points of intersection between the diagonals of a convex octagon
(8-vertex planar polygon)? Note that a polygon is said to be convex if the line segment joining any two points in its interior lies wholly in the interior of the polygon. Only points of intersection between diagonals that lie in the interior of the octagon are to be considered for this problem.
I am not getting how to think about this problem and is there any general solution for such problems?
Best Answer
To explain André Nicolas's comment, for any group of $4$ vertices, there is one interior intersection of the diagonals. Thus, for each of the $\binom{n}{4}$ choice of $4$ vertices, there is one intersection. Therefore, the maximum number of intersections (when there are no coincident intersections) is $$ \binom{n}{4} $$