Question

If $P(n,4)=20\times P(n,2)$, find $n$.

Answer 1

$P(n,4)=20\times P(n,2)$

$\frac{n!}{(n-4)!}=20\times \frac{n!}{(n-2)!}$

$(n-2)!=20\times (n-4)!$

$(n-2)(n-3)(n-4)!=20\times (n-4)!$

$(n-2)(n-3)=20$

$(n-2)(n-3)=5\times 4$

$n^2-5n-14$

$n^2-7n+2n-14$

$(n-7)(n+2)=0\Rightarrow n=7,-2$

$\therefore n=7$ as $n$ cannot be negative