Question 601005
The total number of triangles we can form is *[tex \LARGE {{n} \choose {3}}]. Here, we can subtract the triangles that have exactly one diagonal as an edge.* For this to occur, we need two edges of the triangle on the perimeter of the n-gon, which occurs if and only if all three vertices are consecutive. The number of such triangles is simply n (because we can fix a middle vertex, there are n vertices).


Therefore the number of triangles with more than one diagonal as an edge is


*[tex \LARGE {{n} \choose {3}} - n = \frac{n(n-1)(n-2) - 6n}{6}]


*We cannot have any triangles with all three edges on the perimeter.