Question

Let $t_n$ be the number of possible quadrilaterals formed by joining vertices of an n sided regular polygon. If $t_{n+1}-t_n=56$, then the number of triangles formed by joining vertices of the polygon is

• A. $120$
• B. $84$
• C. $56$
• D. $35$

Answer 1

Answer: (C)

As $t_n$ is the number of ways of selecting four vertices from n sided regular polygon
$\therefore t_n={}^n{C}_4\therefore t_{n+1}-t_n=56$
$\Rightarrow {}^{n+1}{C}_4-{}^n{C}_4=56$
$\Rightarrow {}^n{C}_4+{}^n{C}_3-{}^n{C}_4=56$
$\Rightarrow {}^n{C}_3=56$
$\Rightarrow \frac{n\left(n-1\right)\left(n-2\right)}{3\cdot 2\cdot 1}=7\times 8$
$\Rightarrow n\left(n-1\right)\left(n-2\right)=8\cdot 7\cdot 6$
$\Rightarrow n=8$
$\therefore$ Required number of triangles
$={}^n{C}_3={}^8{C}_3=\frac{8\cdot 7\cdot 6}{3\cdot 2\cdot 1}=56$