Question

If ${}^n{C}_8{=}^n{C}_2$, find ${}^n{C}_2$.

Answer 1

${}^n{C}_8{=}^n{C}_2$

${}^n{C}_r{=}^n{C}_p$ is true only when $r=p$ or $n-r=p$

First case is not possible hence $n-r=p$

Here $r=8$ and $p=2$

$n-r=p$

$n-8=2$

$n=10$

${}^{10}{C}_2=\frac{10!}{8!\times 2!}=\frac{10\times 9\times 8!}{8!\times 2!}=\frac{10\times 9}{2}=45$

Hence ${}^{10}{C}_2=45$