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

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Solution

$\frac{^{n} P_{3}}{^{n} P_{6}} = \frac{1}{210}$
$\Rightarrow \frac{\frac{n!}{(n - 3)!}}{\frac{n!}{(n - 6)!}} = \frac{1}{210}$
$\Rightarrow \frac{(n - 6)!}{(n - 3)!} = \frac{1}{(n - 3) (n - 4) (n - 5)} = \frac{1}{5 \times 6 \times 7}$
$\Rightarrow \frac{1}{(n - 3) (n - 4) (n - 5)} = \frac{1}{7 \times 6 \times 5}$
So $n - 3 = 7$ $n - 4 = 6$ $n - 5 = 5$
$n = 10$ $n = 10$ $n = 10$
So $n = 10$ .

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