Question

If $10\times {}^n{C}_2=3\times {}^{(n+1)}{C}_3$, find $n$.

Answer 1

We have, $10\times {}^n{C}_2=3\times {}^{n+1}{C}_3$

$\Rightarrow 10\times \frac{n!}{(n-2)!2!}=3\times \frac{(n+1)!}{(n-2)!3!}$

$\Rightarrow 10\times \frac{n(n-1)}{2}=3\times \frac{(n+1)\left(n\right)(n-1)}{6}$

$\Rightarrow 10=n+1$, cancel common terms from both sides

$\Rightarrow n=9$

Fact: ${}^n{C}_r=\frac{n!}{(n-r)!r!}$