Question

If ${}^n{C}_3{:}^{2n-1}{C}_2=$ $8:15$, then $n$ $=$

• A. $5$
• B. $6$
• C. $7$
• D. $8$

Answer 1

The correct option is D $8$

${}^n{C}_3{:}^{2n-1}{C}_2=8:15$
$\Rightarrow 15{\cdot }^n{C}_3=8{\cdot }^{2n-1}{C}_2$
$\Rightarrow 15\cdot \frac{n(n-1)(n-2)}{3!}=8\cdot \frac{(2n-1)(2n-2)}{2!}$
$\Rightarrow 15\cdot \frac{n(n-1)(n-2)}{6}=8\cdot \frac{2(2n-1)(n-1)}{2}$
$\Rightarrow 5\cdot \frac{n(n-2)}{2}=8\cdot \frac{(2n-1)}{1}$
$\Rightarrow 5n(n-2)=16(2n-1)$
$\Rightarrow 5n^2-42n+16=0$

$\Rightarrow n=8$
Note: Easiest way will be is to put values of ${}^{\prime }{n}^{\prime }$ and check options