Question

How many diagonals are there in a polygon with $n$ sides?

Answer 1

A polygon of n sides has n vertices. By joining any two vertices of a polygon, we obtain either a side or a diagonal of the polygon. Number of line segments obtained by joining the vertices of a n sided polygon taken two at a time.
= Number of ways of selecting 2 out of n
${=}^n{C}_2=\frac{n(n-1)}{2}$
Out of these lines, n lines are the sides of the polygon.
$\therefore$ Number of diagonals of the polygon
$=\frac{n(n-1)}{2}-n=\frac{n(n-3)}{2}$