Question

If ${}^{10}{P}_r=5040,$ Then what is the value of $r.$

• A. $r=6$
• B. $r=10$
• C. $r=4$
• D. $r=8$

Answer 1

The correct option is C $r=4$

We know that ${}^n{P}_r=\frac{n!}{(n-r)!}.$
In the above question, it is given that $n=10$ and we have to find $r$.

Now, according to the question,
${}^{10}{P}_r=5040$

$\Rightarrow \frac{10!}{(10-r)!}=5040$

$\Rightarrow \frac{10!}{(10-r)!}=10\times 9\times 8\times 7$

$\Rightarrow \frac{10\times 9\times 8\times 7\times 6!}{(10-r)!}=10\times 9\times 8\times 7$

$\Rightarrow \frac{6!}{(10-r)!}=1$

$\Rightarrow 6!=(10-r)!$

$\Rightarrow 6=10-r$

$\Rightarrow r=10-6=4$