# Expected Triangle In Regular N-gon

Calculus Level 2

Consider the $$N$$ vertices of a regular $$N-$$gon that is inscribed in a unit circle. There are $${ N \choose 3 }$$ possible subsets of 3 vertices, which form triangles. If the expected area of the triangle is

$\frac{ 5 \sqrt{ 5 + 2 \sqrt{5} } } { 24 } ,$

what is the value of $$N$$?

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