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