Question

Find n if ${}^{n-1}{P}_3{:}^n{P}_4=1:9$.

Answer 1

Given, ${}^{n-1}{P}_3{:}^n{P}_4=1:9$

$\Rightarrow \frac{{}^{n-1}{P}_3}{{}^n{P}_4}=\frac{1}{9}$

$\Rightarrow \frac{\frac{(n-1)!}{(n-1-3)!}}{\frac{n!}{(n-4)!}}=\frac{1}{9}$

$\Rightarrow \frac{(n-1)!}{(n-4)!}\cdot \frac{(n-4)!}{n!}=\frac{1}{9}$

$\Rightarrow \frac{1}{n}=\frac{1}{9}$

$\Rightarrow n=9$