Question

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

• A. 5
• B. 10
• C. 8
• D. 7

Answer 1

Answer: (A)

$1^{st}$ solution : ${}^{n+1}{C}_3{-}^n{C}_3=10$
$\Rightarrow \frac{\left(n+1\right)n\left(n-1\right)}{6}-\frac{n\left(n-1\right)\left(n-2\right)}{6}=10$
$\Rightarrow 3n\left(n-1\right)=60\Rightarrow n\left(n-1\right)=20$
$\Rightarrow n^2-n-20=0$
$\Rightarrow \left(n-5\right)\left(n+4\right)=0\therefore n=5$
$2^{nd}$ solution : ${}^{n+1}{C}_3-{}^n{C}_3=10$
$\Rightarrow {}^n{C}_2=10\Rightarrow \frac{n\left(n-1\right)}{2}=10$
$\Rightarrow n^2-n-20=0.\therefore n=5$
Here we have used ${}^n{C}_r+{}^n{C}_{r+1}={}^{n+1}{C}_{r+1}$