Question

Find $n$, if ${}^{2n}{C}_3{:}^n{C}_2=52:3$

Answer 1

Find n, if ${}^{2n}{C}_3$ ${:}^n{C}_2=52:3$

$\frac{{}^{2n}{C}_3}{{}^n{C}_2}=\frac{52}{3}$

$\therefore \frac{\frac{2n(2n-1)(2n-2)}{3\times 2\times 1}}{\frac{n(n-1)}{2}}=\frac{52}{3}$

$\therefore \frac{2n(2n-1)2(n-1)}{3\times 2\times 1\times n(n-1)}\times 2=\frac{52}{3}$

$\therefore \frac{4(2n-1)}{3}=\frac{52}{3}$

$\therefore 4(2n-1)=52$

$\therefore 2n-1=\frac{52}{4}$

$\therefore 2n-1=13$

$\therefore 2n=13+1$

$\therefore 2n=14$

$\therefore n=7$