Question

Determine $n$, if ${}^{2n}{C}_2{:}^n{C}_2:=9:2$

• A. $5$
• B. $4$
• C. $3$
• D. $2$

Answer 1

Answer: (A)

Given, ${}^{20}{C}_2{:}^n{C}_2=9:2$
$\Rightarrow$ $\frac{{}^{20}{C}_2}{{}^n{C}_2}=\frac{9}{2}$
$\Rightarrow$ $\frac{(2n)!}{2!(2n-2)!}\times \frac{2!(n-2)!}{n!}=\frac{9}{2}$
$\Rightarrow$ $\frac{(2n)(2n-1)(2n-2)!}{2(2n-2)!}\times \frac{2(n-2)!}{n(n-1)(n-2)!}=\frac{9}{2}$
$\Rightarrow$ $\frac{2(2n-1)}{n-1}=\frac{9}{2}$
$\Rightarrow$ $4(2n-1)=9(n-1)$
$\Rightarrow$ $8n-4=9n-9$
$\Rightarrow$ $9n-8n=-4+9$
$\Rightarrow$ $n=5$