If ${16.}^{n} P_{3} = {13.}^{n + 1} P_{3}$ then find $n$ .

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Solution

$16^{n} P_{3} = 13^{n + 1} P_{3}$
$\Rightarrow \frac{16 n!}{(n - 3)!} = \frac{13 (n + 1)!}{(n - 2)!}$
$\Rightarrow 16 = \frac{13 (n + 1)}{n - 2}$
$\Rightarrow 16 n - 32 = 13 n + 13$
$\Rightarrow 3 n = 45$
$\Rightarrow n = 15$

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