Question

If ${}_{}^{2 n}C_{3}^{}:{}_{}^{n}C_{3}^{}=11:1$ , then find the value of $n.$

Answer 1

It is given that, ${}_{}^{2 n}C_{3}^{}:{}_{}^{n}C_{3}^{}=11:1$
$\text{⇒}\frac{2 n \left(2 n - 1\right) \left(2 n - 2\right)}{1 .2.3}\div\frac{n \left(n - 1\right) \left(n - 2\right)}{1 .2.3}=\frac{11}{1}$
$\text{⇒}\frac{4 n \left(n - 1\right) \left(2 n - 1\right)}{6}\times \frac{6}{n \left(n - 1\right) \left(n - 2\right)}=\frac{11}{1}$
$\Rightarrow \frac{4 \left(2 n - 1\right)}{n - 2}=11$
$\Rightarrow 4\left(\right.2n-1\left.\right)=11\left(\right.n-2\left.\right)$
$\Rightarrow 8n-4=11n-22$
$\Rightarrow 3n=18$
$\Rightarrow n=6$