If $12 P r =^{11} P_{6} + 6^{11} P_{5}$ , then $r =$

6

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5

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7

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None of these

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Solution

The correct option is C $7$
Given that:
$^{12} P_{r} =^{11} P_{r} + 6^{11} P_{r}$

$\frac{12!}{(12 - r)!} = \frac{11!}{(11 - r)!} + 6 \frac{11!}{(11 - r)!}$

$\frac{12 \times 11!}{(12 - r) (12 - r - 1)!} = \frac{11!}{(11 - r)!} + 6 \frac{11!}{(11 - r)!}$

$12 = (1 + 6) (12 - r)$
$84 - 7 r = 12$
$7 r = 72$
$r \approx 7$

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