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 $(10,5)$

${}^n{P}_r=30240$ ${}^n{C}_r=252$

$\frac{n!}{(n-r)!}=30240$ ____ (1) $\frac{n!}{(n-r)!r!}=252$_____ (2)

From (1) & (2)

$252\times r!=30240$

$r!=120$

$\Rightarrow r=5$

$\frac{n!}{(n-5)!}=30240$

$n(n-1)(n-2)(n-3)(n-4)=30240$

$\Rightarrow n=10$

$\therefore n,r=(10,5)$