If $^{n} P_{3} :^{n} P_{6} = 1 : 210$ , find $n$ .

10

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4

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5

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9

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Solution

The correct option is B $10$
$^{n} P_{3} :^{n} P_{6} = 1 : 210$
$\frac{\frac{n!}{(n - 3)!}}{\frac{n!}{(n - 6)!}} = \frac{1}{210}$
$\frac{(n - 6)!}{(n - 3)!} = \frac{1}{210}$
$\frac{1}{(n - 3) (n - 4) (n - 5)} = \frac{1}{7 \times 6 \times 5}$
$n - 3 = 7$
$n = 10$
$∴ n = 10$

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