If P (n, 5) : P (n, 3) = 2 : 1, find n.

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Solution

We have, P (n, 5):P (n, 3) = 2:1

$\Rightarrow \frac{n!}{(n - 5)!} \times \frac{(n - 3)!}{n!} = \frac{2}{1} \Rightarrow \frac{n!}{(n - 5)!} \times \frac{(n - 3) (n - 4) (n - 5)!}{n!} = \frac{2}{1} \Rightarrow (n - 3) (n - 4) = 2 \Rightarrow (n - 3) (n - 4) = 2 \times 1 Thus, on comparing the LHS and the RHS in above expression, we get, n - 3 = 2 \Rightarrow n = 5$

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