Question

If ${}^{10}P_r=5040$ . Then, find out $r$.

Answer 1

Solution.

${}^{n}P_r$ is computed as,

${}^{n}P_r=\frac{n!}{(n-r)!}$

$$\begin{gathered} \because {}^{10}P_r=5040 \\ \therefore \frac{10!}{(10-r)!}=5040 \\ \Rightarrow (10-r)!=\frac{10!}{5040} \\ \Rightarrow (10-r)!=\frac{10\times 9\times 8\times 7\times 6!}{5040} \\ \Rightarrow (10-r)!=\frac{5040\times 6!}{5040} \\ \Rightarrow (10-r)!=6! \\ \Rightarrow 10-r=6 \\ \Rightarrow r=10-6 \\ \Rightarrow r=4 \end{gathered}$$

Hence, the value of $r$ is $4$.