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

Answer: (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]
$\Rightarrow \,{}^nC_2 + \,{}^n C_3 - \,{}^nC_3 =10 [\because \,{}^nC_r + {}^nC_{r+1}= \,{}^{n+1}C_{r+1}]$
$\Rightarrow \,{}^nC_2 =10$
$\Rightarrow \, n=5$