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Question
If P (n, 5) : P (n, 3) = 2 : 1, find n.
Solution
We have, P (n, 5):P (n, 3) = 2:1
\[\Rightarrow \frac{n!}{\left( n - 5 \right)!} \times \frac{\left( n - 3 \right)!}{n!} = \frac{2}{1}\]
\[ \Rightarrow \frac{n!}{\left( n - 5 \right)!} \times \frac{\left( n - 3 \right)\left( n - 4 \right)\left( n - 5 \right)!}{n!} = \frac{2}{1}\]
\[ \Rightarrow \left( n - 3 \right)\left( n - 4 \right) = 2\]
\[ \Rightarrow \left( n - 3 \right)\left( n - 4 \right) = 2 \times 1\]
\[\text{Thus, on comparing the LHS and the RHS in above expression, we get}, \]
\[n - 3 = 2\]
\[ \Rightarrow n = 5 \]
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