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

Question

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

✓

Answer

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

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