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I'm having troubles solving the following problem which is about combinatorics: let $n$ be a natural number $\ge 3$, and a convex polygon with $n$ vertices.

Each vertices are supposed to connect each other in straight lines, so that three lines never intersect at the same point.

The problem is about finding the number of intersections inside the polygon with respect to $n$, the number of vertices.

Here is an example with $n = 6$ http://upload.wikimedia.org/wikipedia/commons/3/37/Polygone-convexe.png

Thanks a lot in advance !!

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  • $\begingroup$ Have you tried drawing the diagram for a triangle, a quadrilateral, and a pentagon, counting the intersections, and seeing if you notice a pattern? $\endgroup$
    – MJD
    Oct 20 '14 at 18:46
  • $\begingroup$ yes I did, but didn't see any :( $\endgroup$
    – Philippe
    Oct 20 '14 at 18:53
  • $\begingroup$ Maybe you could start by trying to find a formula for the number of lines that are intersecting. $\endgroup$
    – MJD
    Oct 20 '14 at 19:33
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Hint: From a point of intersection, you find four of the $n$ vertices by following the two lines in both directions.

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