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

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,

$n-3=2$

$\Rightarrow n=5$