Question

Let $T_n$ be the number of all possible triangles formed by joining vertices of $n$-sided regular polygon. If $T_{n+1}-T_n=10$, then the value of $n$ is :

• A. $7$
• B. $5$
• C. $10$
• D. $8$

Answer 1

The correct option is B $5$

$T_n={}^n{C}_3$
$T_{n+1}={}^{n+1}{C}_3$
Since, $T_{n+1}-T_n=10$
$\Rightarrow {}^{n+1}{C}_3-{}^n{C}_3=10$
$\because {}^n{C}_r+{}^n{C}_{r-1}={}^{n+1}{C}_r$
$\therefore {}^n{C}_3+{}^n{C}_2-{}^n{C}_3=10$
$\Rightarrow \frac{n!}{2!(n-2)!}=10$
$\Rightarrow n^2-n-20=0$
$\Rightarrow (n-5)(n+4)=0$
$\Rightarrow n=5$ and $n=-4$
$n=-4$ is not possible
Hence, $n=5$