<p>Twenty distinct points are marked on a circle and labeled <img alt="$1$" height="14" src="https://latex.artofproblemsolving.com/d/c/e/dce34f4dfb2406144304ad0d6106c5382ddd1446.png" width="11" /> through <img alt="$20$" height="12" src="https://latex.artofproblemsolving.com/f/8/3/f8366cd6196dd8a9da6d38a3e9eafb109e99d53e.png" width="17" /> in clockwise order. A line segment is drawn between every pair of points whose labels differ by a prime number. Find the number of triangles formed whose vertices are among the original <img alt="$20$" height="12" src="https://latex.artofproblemsolving.com/f/8/3/f8366cd6196dd8a9da6d38a3e9eafb109e99d53e.png" width="17" /> points.</p>

Question No: 371

Twenty distinct points are marked on a circle and labeled $1$ through $20$ in clockwise order. A line segment is drawn between every pair of points whose labels differ by a prime number. Find the number of triangles formed whose vertices are among the original $20$ points.

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Tags: Combinatorics

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