Question

If ${}^9{P}_5{.}^9{P}_4{=}^{10}{P}_r.$ Find $r$.

Answer 1

We have
${}^9{P}_5{.}^9{P}_4{=}^{10}{P}_r.$
$\Rightarrow \frac{9!}{(9-5)!}+5\frac{9!}{(9-4)!}=\frac{10!}{(10-r)!}$
$\Rightarrow \frac{9!}{4!}+5\frac{9!}{5!}=\frac{10!}{(10-r)!}$
$\Rightarrow \frac{9!}{4!}+\frac{9!}{4!}=\frac{10!}{(10-r)!}$
$\Rightarrow 2\times \frac{9!}{4!}=\frac{10!}{(10-r)!}$
$\Rightarrow \frac{5\times 2\times 9!}{5\times 4!}=\frac{10!}{(10-r)!}$
$\Rightarrow \frac{10\times 9!}{5!}=\frac{10!}{(10-r)!}$
$\Rightarrow \frac{10!}{5!}=\frac{10!}{(10-r)!}$
$\Rightarrow (10-r)!=5!\Rightarrow 10-r=5\Rightarrow r=5$