Question

A regular polygon has $20$ sides. How many triangles can be drawn by using the vertices but not using the sides$?$

Answer 1

The first vertex can be selected in $20$ ways. The remaining two are to be selected from $17$ vertices so that they are not consecutive. This can be done in ${}^{17}{\text{C}}_2-16$ ways. The total number of ways$=20\times ({}^{17}{\text{C}}_2-16)$
But in this method, each selection is repeated thrice.
Number of triangles$=\frac{20\times ({}^{17}{\text{C}}_2-16)}{3}=800.$