Question

Determine n if

i) ${}^{2n}{C}_3{:}^n{C}_3=12:1$

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

Answer 1

i) ${}^{2n}{C}_3{:}^n{C}_3=12:1$
${\Rightarrow }^{2n}{C}_3=12{\times }^n{C}_3$
$\Rightarrow \frac{2n(2n-1)(2n-2)}{3!}$

$=12\times \frac{n(n-1)(n-2)}{3!}$
$\Rightarrow 4(2n-1)=12(n-2)$

$\Rightarrow 2n-1=3(n-2)$
$\Rightarrow 2n-1=3n-6$

$\Rightarrow n=6-1=5$

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

${\Rightarrow }^{2n}{C}_3=11{\times }^n{C}_3$
$\Rightarrow \frac{2n(2n-1)(2n-2)}{3!}$

$=11\times \frac{n(n-1)(n-2)}{3!}$
$\Rightarrow 4(2n-1)=11(n-2)$

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

$\Rightarrow 3n=18$

$\Rightarrow n=6$