Question

If ${}^n{P}_5:{}^{n-1}{P}_5=3:2$, find $n$.

Answer 1

Fact: ${}^n{P}_r=\frac{n!}{(n-r)!}$

We have, ${{}^{n}P}_5:{{}^{n-1}P}_5=3:2$

$\Rightarrow \frac{{}^n{P}_5}{{}^{n-1}{P}_5}=\frac{3}{2}$

$\Rightarrow \frac{\frac{n!}{(n-5)!}}{\frac{(n-1)!}{(n-6)!}}=\frac{3}{2}$

$\Rightarrow \frac{n}{n-5}=\frac{3}{2}$

$\Rightarrow 2n=3n-15\Rightarrow n=15$