Question

T${}_m$ denotes the number of triangles that can be formed with the vertices of a regular polygon of $m$ sides. If $T_{m+1}-T_m=15,$ then $m$ is:

• A. $3$
• B. $6$
• C. $9$
• D. $12$

Answer 1

The correct option is B $6$

$T_m$ denotes number of triangles that can be formed with the vertices of a regular polygon of $m$ sides.

Given,

$T_{m+1}-T_m=15$

${\Rightarrow }^{m+1}{C}_3{-}^m{C}_3=15$

$\Rightarrow \frac{(m+1)!}{(m-2)!.3!}-\frac{m!}{(m-3)!.3!}=15$

$\Rightarrow \frac{m!}{(m-3)!}[\frac{m+1}{m-2}-1]=90$

$\Rightarrow \frac{3!\left(3\right)}{(m-2)!}=90$

$\Rightarrow \frac{m!}{(m-2)!}=30=6.5=\frac{6!}{4!}$

$\Rightarrow m=6$ is the correct answer.