Question

If P(n,5)=20.P(n,3), find n.

Answer 1

We have,
P (n,5) = 20 P(n,3)
$\Rightarrow \frac{n!}{(n-5)!}=20\times \frac{n!}{(n-3)!}$
$\Rightarrow \frac{1}{(n-5)!}=\frac{20}{(n-3)(n-3-1)(n-3-2)!}$
$\Rightarrow \frac{1}{(n-5)!}=\frac{20}{(n-3)(n-3-1)(n-3-2)!}$
$\Rightarrow \frac{1}{(n-5)!}=\frac{20}{(n-3)(n-4)(n-5)!}$
$\Rightarrow \frac{(n-3)(n-4)(n-5)!}{(n-5)!}=20$
$\Rightarrow (n-3)(n-4)=20$
$\Rightarrow n^2-4n-3n+12=20$
$\Rightarrow n^2-8n+1n-8=0$
$\Rightarrow n(n-8)+1(n-8)=0$
$\Rightarrow n-8=0[\because n\ne -1]$
$\Rightarrow n=8$
Hence, n=8