Question

The number of diagonals of a polygon of $m$ sides is?

• A. $\frac{1}{2!}\left[m\right(m-5\left)\right]$
• B. $\frac{1}{2!}\left[m\right(m-1\left)\right]$
• C. $\frac{1}{2!}\left[m\right(m-3\left)\right]$
• D. $\frac{1}{2!}\left[m\right(m-2\left)\right]$

Answer 1

The correct option is C

$\frac{1}{2!}\left[m\right(m-3\left)\right]$

Explanation for the correct option:

Step 1: Find the required number of diagonals.

We know that the total number of lines formed through $m$ vertices is given by ${}^{m}C_2$.

Therefore, the total number of lines formed by vertices of an $m$ sided polygon is ${}^{m}C_2=\frac{m(m-1)}{2!}$.

So, the number of diagonals is given by $\frac{m(m-1)}{2!}-1=\frac{m(m-3)}{2!}$.

Therefore, the number of diagonals in a polygon of $m$ sides is $\frac{1}{2!}\left[m\right(m-3\left)\right]$.

Hence, option $C$ is the correct answer.