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 $^{n}C_{r}+\,{}^{n}C_{r+1}=\,{}^{n+1}C_{r+1}$