Question:
If P (n − 1, 3) : P (n, 4) = 1 : 9, find n.
Solution:
P (n − 1, 3):P (n, 4) = 1:9
$\Rightarrow \frac{(n-1) !}{(n-1-3) !} \times \frac{(n-4) !}{(n) !}=\frac{1}{9}$
$\Rightarrow \frac{(n-1) !}{(n-4) !} \times \frac{(n-4) !}{n !}=\frac{1}{9}$
$\Rightarrow \frac{(n-1) !}{n !}=\frac{1}{9}$
$\Rightarrow \frac{(n-1) !}{n(n-1) !}=\frac{1}{9}$
$\Rightarrow \frac{1}{n}=\frac{1}{9}$
$\Rightarrow n=9$
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