If $^{56} P_{r + 6} :^{54} P_{r + 3} = 30800$ , then $r$ is

39

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41

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28

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43

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Solution

The correct option is B $41$
$^{56} P_{r + 6} :^{54} P_{r + 3} = 30800$
$\frac{\frac{56!}{(50 - r)!}}{\frac{54!}{(51 - r)!}} = 30800$
$\frac{56! \times (51 - r)!}{54! (50 - r)!} = 30800$
$56 \times 55 \times (51 - r) = 30800$
$(51 - r) = \frac{30800}{56 \times 55}$
$(51 - r) = 10$
$r = 41$

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