Consider a 20-sided convex polygon K, with vertices \[A_1, A_2, . . . , A_{20} \] in that order. Find the number of ways in which three sides of K can be chosen so that every pair among them has at least two sides of K between them.

**Clarification**:

For example \[(A_1 A_2, A_4 A_5, A_{11} A_{12})\] is an admissible triple while \[(A_1 A_2, A_4 A_5, A_{19} A_{20})\] is not.

The language of the question is correct. The question is a modified form of one that appeared in CRMO 2011.

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