Crossing chords

Task number: 3458

In a convex \(n\)-gon, how many pairs of chords exist that intersect inside the \(n\)-gon, i.e. not at boundary points?

  • Solution

    Every pair of chords gives us a quadruple of vertices, the endpoints of the chords.

    Conversely, if we choose a quadruple of vertices, we can then choose only one pair of chords that will intersect. (With the other two possibilities, the chords do not intersect.)

  • Answer

    The number of pairs of chords is \(\binom{n}{4}\).

Difficulty level: Hard task
Solution require uncommon idea
Cs translation
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