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 2 of 6: Count the five-point subsets
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#{5-point subsets}=(n5)\#\{\text{5-point subsets}\}=\binom n5
Detailed analysis

There are (n5)\binom n5 choices of five given points, and each choice supplies at least one convex quadrilateral.