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 4 of 6: Obtain a first lower bound
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Q≥(n5)n−4=n(n−1)(n−2)(n−3)120Q\ge\frac{\binom n5}{n-4}=\frac{n(n-1)(n-2)(n-3)}{120}
Detailed analysis

If QQ is the number of convex quadrilaterals, double counting the pairs (five-point subset, selected quadrilateral) gives the displayed inequality.