MathLabs

Problem 5

Given n>4n>4 points in the plane, no three collinear, prove that at least (n−32)\binom{n-3}{2} convex quadrilaterals have their vertices among the given points.
Step 3 of 6: Control overcounting
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#{5-subsets containing a fixed quadrilateral}=n−4\#\{\text{5-subsets containing a fixed quadrilateral}\}=n-4
Detailed analysis

A fixed convex quadrilateral uses four points, and its containing five-point subset is obtained by choosing one of the remaining n−4n-4 points. Hence every quadrilateral was counted at most n−4n-4 times.