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Question
If P (n, 4) = 12 . P (n, 2), find n.
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Solution
P (n, 4) = 12 . P (n, 2)
\[\Rightarrow \frac{n!}{\left( n - 4 \right)!} = 12 \times \frac{n!}{\left( n - 2 \right)!}\]
\[ \Rightarrow \frac{\left( n - 2 \right)!}{\left( n - 4 \right)!} = 12 \times \frac{n!}{n!}\]
\[ \Rightarrow \frac{\left( n - 2 \right)\left( n - 3 \right)\left( n - 4 \right)!}{\left( n - 4 \right)!} = 12\]
\[ \Rightarrow \left( n - 2 \right)\left( n - 3 \right) = 12\]
\[ \Rightarrow \left( n - 2 \right)\left( n - 3 \right) = 4 \times 3\]
\[\text{On comparing the LHS and the RHS, we get}: \]
\[n - 2 = 4\]
\[ \Rightarrow n = 6\]
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