The coefficient of x^ - n in (1 + x)n(1 + (1)/(x))n is

\left(1 + x\right)n\left(1 + \dfrac{1}{x}\right)n = \left[\left(1 + x\right)\left(1 + \dfrac{1}{x}\right)\right]n$</p><p>\left(1 + x\right)n\left(1 + \dfrac{1}{x}\right)n = \left[1 + x + \dfrac{1}{x} + 1\right]n$

\left(1 + x\right)n\left(1 + \dfrac{1}{x}\right)n = \left[\left(2 + x\right) + \dfrac{1}{x}\right]n$</p><p>\left(1 + x\right)n\left(1 + \dfrac{1}{x}\right)n = C0 n \left(2 + x\right)n \cdot \left(\dfrac{1}{x}\right) + C1 n \left(2 + x\right)n - 1 \cdot \left(\dfrac{1}{x}\right)1 + \ldots + Cn n \cdot \dfrac{1}{xn}$

$∴$ Coefficient of $x^{- n} = Cn n = 1$