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Question
If n +5Pn +1 =\[\frac{11 (n - 1)}{2}\]n +3Pn, find n.
Solution
n +5Pn +1 =11(n-1)2n +3Pn
\[ \Rightarrow \frac{\left( n + 5 \right)!}{4!} = \frac{11\left( n - 1 \right)}{2} \times \frac{\left( n + 3 \right)!}{3!}\]
\[ \Rightarrow \frac{\left( n + 5 \right)!}{\left( n + 3 \right)!} = \frac{11\left( n - 1 \right)}{2} \times \frac{4!}{3!}\]
\[ \Rightarrow \frac{\left( n + 5 \right)\left( n + 4 \right)\left( n + 3 \right)!}{\left( n + 3 \right)!} = \frac{11\left( n - 1 \right)}{2} \times \frac{4 \times 3!}{3!}\]
\[ \Rightarrow \left( n + 5 \right)\left( n + 4 \right) = 22\left( n - 1 \right)\]
\[ \Rightarrow n^2 + 9n + 20 = 22n - 22\]
\[ \Rightarrow n^2 - 13n + 42 = 0\]
\[ \Rightarrow n = 7, 6\]
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