Question

Let $T_n$ be the number of triangles of which can be formed using the vertices of a regular polygon of $n$ sides. If $T_{n+1}+T_n=91$ , then $n$ equals:

• A. $8$
• B. $7$
• C. $13$
• D. $9$

Answer 1

The correct option is B $7$

${}^{n+1}{C}_3{+}^n{C}_3=91n+1{(}^n{C}_3\left){+}^n{C}_3{}^{}{}^n{C}_3\right(n+2)=91\frac{n!}{(n-3)!3!}(n+2)=91\frac{n(n-1)(n-2)(n+2)}{3!}91\Rightarrow n=7$