Question

Determine $n$ if, ${}^{2n}{C}_3{:}^n{C}_3=12:1$

Answer 1

Determine n, if ${}^{2n}{C}_3{:}^n{C}_3=12:1$

${}^{2n}{C}_3={12}^n{C}_3$

$\frac{2n!}{2n-3!3!}=12\frac{n!}{n-3!3!}$

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

$\Rightarrow \frac{n(2n-1)(n-1)}{3}=2\left(n\right)(n-1)(n-2)$

$2n-1=3n-6$

$tn=t5$

$n=5$