If ${}^{2n}C_3 : {}^nC_2 = 44 : 3$, then the value of $n$ is–

Explanation

$$ \frac{{}^{2n}C_3}{{}^nC_2} = \frac{44}{3} $$ $$ \frac{\frac{2n}{n-2 \times 2}}{\frac{2n-3 \times 3}{n}} = \frac{44}{3} $$ $$ \frac{2n \times (2n-1)(2n-2)}{3 \times 2 \times 1} \div \frac{n(n-1)}{2 \times 1} = \frac{44}{3} $$ $$ \frac{2n \times 2 \times (2n-1)(n-1)}{3n(n-1)} = \frac{44}{3} $$ $$ \frac{4(2n-1)}{3} = \frac{44}{3} $$ $$ 2n - 1 = 11 \\ 2n = 12 \\ \therefore n = 6 $$