Crossing of the Chords

Crossing of the Chords

A number of points (N) are distributed along the circumference of a circle. Every point is connected to every other point by a chord. The points are spaced unevenly such that no more than two chords intersect at a common point inside the circle.

   The October Geek Challenge involves some geometry.

As demonstrated in these figures, when there are 4, 5 and 6 perimeter points, there are 1, 5 and 15 internal chord intersections respectively.

If N=8 (circle with 8 points on the perimeter), how many internal cord intersections are there?

A:  56
B:  64
C:  70
D:  85

Extra Credit: Produce a simplified expression that represents the number of internal chord intersections as a function of the number of perimeter points (N).

As always, the answer with the best analytical content will be this month’s Geek Challenge winner!

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