If ^ 22 P_ r+1 : ^ 20 P_ r+2 =11: 52. find r . - Vidyadip

Question

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

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Answer

Here ${ }^{22} P_{r+1}:{ }^{20} P_{r+2}=11: 52$
$\Rightarrow \frac{22!}{(21-r)!} \times \frac{(18-r)!}{20!}=\frac{11}{52}$
$\Rightarrow=\frac{22 \times 21 \times 20!}{(21-r)(20-r)(19-r)(18-r)!} \times \frac{(18-r)!}{20!}=\frac{11}{52}$
$\Rightarrow \frac{22 \times 21}{(21-r)(20-r)(19-r)}=\frac{11}{52}$
$\Rightarrow(21-r)(20-r)(19-r)=2 \times 21 \times 52$
$\Rightarrow(21-r)(20-r)(19-r)=14 \times 13 \times 12$
$\Rightarrow(21-r)(20-r)(19-r)=(21-7)(20-7)(19-7)$
$\Rightarrow r=7$

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