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If ${ }^{56} P_{r+6}:{ }^{54} P_{r+3}=30800: 1$, then
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Verified Answer
The correct answer is:
$r=41$
Given, ${ }^{56} P_{r+6}:^{54} P_{r+3}=30800: 1$
$\Rightarrow \quad \frac{{ }^{56} P_{r+6}}{{ }^{54} P_{r+3}}=\frac{30800}{1}$
$\Rightarrow \quad \frac{56 ! \times(51-r) !}{(50-r) ! \times 54 !}=\frac{30800}{1}$
$\Rightarrow 56 \times 55 \times(51-r)=30800$
$\Rightarrow \quad 51-r=10 \Rightarrow r=41$
$\Rightarrow \quad \frac{{ }^{56} P_{r+6}}{{ }^{54} P_{r+3}}=\frac{30800}{1}$
$\Rightarrow \quad \frac{56 ! \times(51-r) !}{(50-r) ! \times 54 !}=\frac{30800}{1}$
$\Rightarrow 56 \times 55 \times(51-r)=30800$
$\Rightarrow \quad 51-r=10 \Rightarrow r=41$
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