Question

If ${}^9{P}_5+{5.}^9{P}_4{=}^{10}{P}_r$, find r.

Answer 1

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

L.H.S. = $\frac{9!}{4!}+5.\frac{9!}{5!}$, but $\frac{5}{5!}=\frac{1}{4!}$

= $\frac{2.9!}{4!}=\frac{2\times 5.9!}{5\times 4!}=\frac{10.9!}{5!}=\frac{10!}{5!}$

R.H.S.= $\frac{10!}{(10-r)!}=\frac{10!}{5!}$ = L.H.S

$\therefore$ 10 - r = 5 or r = 5.