Question

Determine n if ${}^{2n}{C}_3{:}^n{C}_3=11:1$

Answer 1

Given ${}^{2n}{C}_3{:}^n{C}_3=11:1$

${\Rightarrow }^{2n}{C}_3=11{\times }^n{C}_3$

$\Rightarrow \frac{2n(2n-1)(2n-2)}{3!}=11\times \frac{n(n-1)(n-2)}{3!}$ $[\because {}^n{C}_r=\frac{n!}{(n-r)!\times r!}]$

$\Rightarrow 4(2n-1)=11(n-2)$

$\Rightarrow 8n-4=11n-22$

$\Rightarrow 11n-8n=22-4$

$\Rightarrow 3n=18$

$\Rightarrow n=6$