Let n &gt; 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?Explain your answer, which should be expressed in terms of n.

Question

Let   n  &amp;gt;  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?Explain your answer, which should be expressed in terms of   n.

Madilyn Dunn · Accepted Answer

We need to count the number of ways to choose   4 vertices so that at least two pairs of vertices are adjacent.There are                               (                                      n          −          3                4                              )                      −                              (                                      n          −          5                2                              )                     ways to choose them so that none are adjacent,and there are   n                              (                                      n          −          5                2                              )                     ways to choose them so that exactly two are adjacent;so this gives a total of                               (                            n        4                              )                      −                            (                                      n          −          3                4                              )                      −    (    n    −    1    )                            (                                      n          −          5                2                              )                      =                  n        (        3        n        −        13        )            2       ways to choose the vertices.

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? Explain your answer, which should be expressed in terms of n.

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