# Intersecting Chords

Discrete Mathematics Level 5

Two points are chosen uniformly at random on the unit circle and joined to make a chord $$C_1$$. This process is repeated $$17$$ more times to get chords $$C_2, C_3, \ldots, C_{18}$$. What is the expected number of pairs of chords that intersect?

Details and assumptions

If $$k$$ chords intersect at the same point, this counts as $$\binom{k}{2}$$ pairs of intersecting chords.

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