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

$(1 + x)^{n} (1 + \frac{1}{x})^{n} = \frac{(1 + x)^{n} (1 + x)^{n}}{x^{n}} = \frac{(1 + x)^{2 n}}{x^{n}}$

Therefore, coefficient of $x^{- 1}$ in $(1 + x)^{n} (1 + \frac{1}{x})^{n}$ = Coefficient of $\frac{(1 + x)^{2 n}}{x^{n}}$

$=$ Coefficient of $x^{n - 1}$ in $(1 + x)^{2 n}$

$= 2 n_{C_{n - 1}}$

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