Question

If ${}^n{P}_r=5040$ and ${}^n{C}_r=210$, then find $n$ and $r$.

Answer 1

We have, ${}^n{P}_r=5040$ and ${}^n{C}_r=210$,

$\frac{n!}{(n-r)!}=5040$ and $\frac{n!}{(n-r)!r!}=210$

Divide both

$\Rightarrow r!=24\Rightarrow r=4$

So $\frac{n!}{(n-4)!}=5040$

$\Rightarrow \frac{n(n-1)(n-3)(n-4)!}{(n-4)!}=5040$

$\Rightarrow n(n-1)(n-2)(n-3)=5040$

$\Rightarrow n=10$