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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