If n is a positive integer, find the coefficient of $x^{- 1}$ in the expansion of $(1 + x)^{n} {(1 + \frac{1}{x})}^{n}$

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Solution

We have
$(1 + x)^{n} {(1 + \frac{1}{x})}^{n} = (1 + x)^{n} \times \frac{(1 + x)^{n}}{x^{n}} = \frac{(1 + x)^{2 n}}{x^{n}}$
Coefficient of $x^{- 1}$ in the expansion of $(1 + x)^{n} {(1 + \frac{1}{x})}^{n}$
= coefficient of $x^{- 1}$ in the expansion of $(1 + x)^{n} {(1 + \frac{1}{x})}^{n} = Coefficient of x^{- 1} in the expansion of \frac{(1 + x)^{2 n}}{x^{n}} = Coefficient of (x^{- 1} \times x^{n}) in the expansion of (1 + x)^{2 n} = Coefficient of x^{n - 1} in the expansion of (1 + x)^{2 n} =^{2 n} C_{n - 1}$

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