Question

If ${}^{22}{P}_{r+1}{:}^{20}{P}_{r+2}=11:52,$ find r.

Answer 1

We have

${}^{22}{P}_{r+1}{:}^{20}{P}_{r+2}=11:52,$

$\Rightarrow \frac{22!}{\{22-(r+1\left)\right\}!}:\frac{20!}{\{20-(r+2\left)\right\}!}=11:52$

$\Rightarrow \frac{22!}{(21-r)!}:\frac{20!}{(18-r)!}=\frac{11}{52}$

$\Rightarrow \frac{22!}{(21-r)!}\times \frac{(18-r)!}{20!}=\frac{11}{52}$

$\Rightarrow \frac{22\times 21\times (20!)}{(21-r)(20-r)(19-r)\times [18-r)]!}\times \frac{(18-r)!}{20!}=\frac{11}{52}$

$\Rightarrow \frac{22\times 21}{(21-r)(20-r)(19-r)}=\frac{11}{52}$

$\Rightarrow (19-r)(20-r)(21-r)=2\times 21\times 52=2\times 3\times 7\times 4\times 13$

$\Rightarrow (19-r)(20-r)(21-r)=12\times 13\times 14$

$\Rightarrow 19-r=12\Rightarrow r=(19-12)=7.$

Hence, $r=7$