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. 7
• B. 5
• C. 10
• D. 8

Answer 1

The Correct Option is B

Given , $T_n= \, ^nC_3 \, \Rightarrow \, T_{n+1} = \, ^{n+1}C_3$
$\therefore \, \, \, $ $T_{n+1}-T_n = \, ^{n+1}C_3 - \, ^nC_3 =10 \, \, \, \, \, \, \, \, \, \, \, [given]$
$\Rightarrow \, ^nC_2 + \, ^n C_3 - \, ^nC_3 =10 \, \, \, \, \, \, \, [\because \, ^nC_r \, + \, ^nC_{r+1}= \, ^{n+1}C_{r+1}]$
$\Rightarrow \, ^nC_2 =10$
$\Rightarrow \, n=5$