



 
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 n1 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,  


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