SOLUTION: Let n>4.
In how many ways can we choose 4 vertices of a convex n-gon so as to form a convex quadrilateral, such that at least 2 sides of the quadrilateral are sides of the n-gon
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Question 621807: Let n>4.
In how many ways can we choose 4 vertices of a convex n-gon so as to form a convex quadrilateral, such that at least 2 sides of the quadrilateral are sides of the n-gon?
Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
Label the vertices 
.
We have two cases:
Case 1: Three of the four sides of the quadrilateral are sides of the n-gon.
Case 2: Exactly two of the four sides are sides on the n-gon.
Case 1 is pretty easy to count; all four vertices have to be consecutive. Hence we can have quadrilaterals 
, 
, ..., 
, n quadrilaterals.
Case 2 is a little trickier. The n-gon obviously has n-sides, so the solution is to choose two sides on the n-gon that are not consecutive. This can occur in
}{2})
ways (we must divide by 2 because we are counting each pair of sides twice).
Hence the total number of ways to choose four vertices in this manner is }{2} = \frac{n^2 - n}{2})
, which happens to be 
.
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