Question

If P (n,4) = 12. P (n,2), find n.

Answer 1

We have,
P(n,4) = 12. P(n,2)
$\Rightarrow \frac{n!}{(n-4)!}=12\times \frac{n!}{(n-2)!}\Rightarrow \frac{1}{(n-4)!}=\frac{12}{(n-2)!}\Rightarrow \frac{1}{(n-4)!}=\frac{12}{(n-2)(n-3)(n-4)!}\Rightarrow \frac{(n-2)(n-3)(n-4)}{(n-4)!}=12\Rightarrow (n-2)(n-3)=12\Rightarrow n^2-3n-2n+6=12\Rightarrow n^2-5n+6-12=0\Rightarrow n^2-5n-6=0\Rightarrow n^2-6n+1n-6=0\Rightarrow n(n-6)(n+1)=0\Rightarrow n-6=0[\because n\ne -1]\Rightarrow n=6$
Hence, n= 6