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

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Solution

$16^{n} P_{3} = 13^{n + 1} P_{3} (Given)$

$16 \frac{n!}{(n - 3)!} = 13 \frac{(n + 1)!}{(n + 1 - 3)!} (∵^{n} P_{r} = \frac{n!}{(n - r)!})$

$\Rightarrow 16 \times \frac{n!}{(n - 3)!} = 13 \times \frac{(n + 1) n!}{(n - 2) (n - 3)!}$

$\Rightarrow 16 (n - 2) = 13 (n + 1)$

$\Rightarrow 16 n - 32 = 13 n + 13$

$\Rightarrow 16 n - 13 n = 32 + 13$

$\Rightarrow n = \frac{45}{3} = 15$

Hence value of $n$ is $15$ .

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