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
$\Rightarrow \frac{P (n, 5)}{P (n, 3)} = \frac{2}{1} \Rightarrow \frac{\frac{n!}{(n - 5)!}}{\frac{n!}{(n - 3)!}} = \frac{2}{1} \Rightarrow \frac{n! \times (n - 3)!}{(n - 5)! \times n!} = \frac{2}{1} \Rightarrow \frac{n! \times (n - 3)!}{(n - 5)! \times n!} = 2 \Rightarrow \frac{(n - 3)!}{(n - 5)!} = 2 \Rightarrow \frac{(n - 3) (n - 4) (n - 5)!}{(n - 5)!} = 2 \Rightarrow (n - 3) (n - 4) = 2 \Rightarrow n^{2} - 4 n - 3 n + 12 = 2 \Rightarrow n^{2} - 7 n + 12 - 2 = 0 \Rightarrow n^{2} - 7 n + 10 = 0 \Rightarrow n^{2} - 5 n - 2 n + 10 = 0 \Rightarrow n (n - 5) - 2 (n - 5) = 0 \Rightarrow (n - 5) (n - 2) = 0 \Rightarrow n = 5 [\begin{matrix} ∵ n \geq 5 ∴ n \neq 2 \end{matrix}]$
Hence, n=5

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