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I have a circle with N points on it, and I want to determine how many triangles can be formed using these points.

How can I do this?



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The vertices of those triangles have to be from these $N$ points, right? – Patrick Li Oct 3 '12 at 20:24
@PatrickLi Yes - that is right. – AndyGeek Oct 3 '12 at 20:25
The question is unclear. What's the relevance of the points being on a circle ? Please show an example figure and pinpoint the triangles that should be counted. – Yves Daoust Mar 31 at 6:47

Each set of $3$ of the $N$ points determines a triangle, and each triangle is determined in this way, so all you have to do is determine how many $3$-element subsets a set of $N$ things has. If you don’t already know this, you should read this article.

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That's one way of interpreting the question. Another (more interesting, in my opinion) is to look at all triangles that can be generated by chords using $N$ points on a circle. For example, with $N=4$ points we have, essentially, divided the circle and its interior into 4 triangular regions, defined by a square and its two diagonals. For $N=5$ we'll have 11 possible triangles (look at $K_5$). If you allow overlapping triangles, you get an even larger number. – Rick Decker Oct 4 '12 at 0:41
@Brian M. Scott: Answer would be NC3 which is nothing but N(N-1)(N-2)/6, am I correct? – Unknown Dec 28 '13 at 18:01

The number of triangles that can be formed given N non collinear points is n = N(N-1)(N-2) / 6

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