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Question
The coefficient of \[\frac{1}{x}\] in the expansion of \[\left( 1 + x \right)^n \left( 1 + \frac{1}{x} \right)^n\] is
Options
\[\frac{n !}{\left[ \left( n - 1 \right) ! \left( n + 1 \right) ! \right]}\]
\[\frac{\left( 2n \right) !}{\left[ \left( n - 1 \right) ! \left( n + 1 \right) ! \right]}\]
\[\frac{\left( 2n \right) !}{\left( 2n - 1 \right) ! \left( 2n + 1 \right) !}\]
none of these
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Solution
\[\frac{\left( 2n \right) !}{\left[ \left( n - 1 \right) ! \left( n + 1 \right) ! \right]}\]
\[\text{ Coefficient of } \frac{1}{x}\text{ in the given expansion = Coefficient of 1 in } (1 + x )^n \times \text{ Coefficient of} \frac{1}{x}in \left( 1 + \frac{1}{x} \right)^n + \text{ Coefficient of x in } (1 + x )^n \times \text{ Coefficient of } \frac{1}{x^2} \text{ in } \left( 1 + \frac{1}{x} \right)^n \]
\[ =^{n}{}{C}_0 \times ^{n}{}{C}_1 +^{n}{}{C}_1 \times ^{n}{}{C}_2 \]
\[ = n + n \times \frac{n!}{2\left( n - 2 \right)!}\]
\[ = n + n\frac{n\left( n - 1 \right)}{2}\]
\[ =\] \[\frac{\left( 2n \right) !}{\left[ \left( n - 1 \right) ! \left( n + 1 \right) ! \right]}\]
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