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 5 of 6: Compare with the target
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n(n−1)(n−2)(n−3)120≥(n−3)(n−4)2\frac{n(n-1)(n-2)(n-3)}{120}\ge\frac{(n-3)(n-4)}2
Detailed analysis

After cancelling the positive factor n−3n-3, this is equivalent to n(n−1)(n−2)≥60(n−4)n(n-1)(n-2)\ge60(n-4).