Question

Solve : ${}^9{P}_5+5{}^9{P}_4{=}^{10}{P}_n$

Answer 1

${}^9{P}_5+5{}^9{P}_4={}^{10}{P}_n$

LHS

$\frac{9!}{4!}+\frac{9!}{5!}\times 5$

$\Rightarrow \frac{9!}{4!}+\frac{9!}{5\cdot 4!}\times 5$

$\Rightarrow 2\times \frac{9!}{4!}$

$\Rightarrow \frac{10\times 9!}{5\times 4!}$

$\Rightarrow \frac{10!}{5!}$

$\Rightarrow {}^{10}{P}_5$

RHS

$\Rightarrow {}^{10}{P}_5={}^{10}{P}_n$

$\Rightarrow n=5$.