Question

If ${}^n{P}_r=30240$ and ${}^n{C}_r=252$, then the ordered pair $(n,r)$ is equal to

• A. $(12,6)$
• B. $(10,5)$
• C. $(9,4)$
• D. $(16,7)$

Answer 1

The correct option is B $(10,5)$

Given that, ${}^n{P}_r=30240$ and ${}^n{C}_r=252$
$\Rightarrow \frac{n!}{(n-r)!}=30240$ and $\frac{n!}{(n-r)!r!}=252$
$\Rightarrow r!=\frac{30240}{252}=120$
$\Rightarrow r=5$
$\therefore \frac{n!}{(n-5)!}=30240$
$\Rightarrow n(n-1)(n-2)(n-3)(n-4)=30240$
$\Rightarrow n(n-1)(n-2)(n-3)(n-4)=10(10-1)(10-2)(10-3)(10-4)$
$\Rightarrow n=10$
Hence, required ordered pair is $(10,5)$