Hi Megan,

Let's first look at lines in a polygon with n vertices. If you are going to join two vertices there are n choices for the starting point. Once you have the starting point there are n-1 choices for the end point. Thus it seems you have n(n - 1) possible lines, but you have counted each line twice, once in each direction. Thus the number of lines joining vertices is n(n - 1)/2, exactly what you got.

Now look at triangles. To draw a triangle you first choose a starting point and you have n possible choices. There are then n - 1 choices for the second point and n - 2 choices for the third point. Hence, as for lines, it seems you have n(n - 1)(n - 2) possible triangles. But consider a particular triangle. How many times did you count it? There are 3 way to choose one of its vertices as the first vertex, 2 ways to choose the second vertex and only 1 way to choose the third vertex. Thus each triangle got counted 3 × 2 × 1 times. Hence the triangles joining vertices of a polygon with n vertices is n(n - 1)(n - 2)/(3 × 2 × 1).

I hope this helps,
Penny