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Question
Find the coefficient of:
(v) \[x^m\] in the expansion of \[\left( x + \frac{1}{x} \right)^n\]
Solution
Suppose xm occurs at the (r + 1)th term in the given expression.
Then, we have:
\[T_{r + 1} = ^{n}{}{C}_r x^{n - r} \frac{1}{x^r}\]
\[ = ^{n}{}{C}_r x^{n - 2r} \]
\[\text{ For this term to contain} x^m , \text{ we must have: } \]
\[n - 2r = m\]
\[ \Rightarrow r = (n - m)/2\]
\[ \therefore \text{ Coefficient of } x^m = ^ {n}{}{C}_{(n - m)/2} = \frac{n!}{\left( \frac{n - m}{2} \right)! \left( \frac{n + m}{2} \right)!}\]
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