The coefficient of frac{1}{x} in the expansio | vedclass

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(B) Given expression: $(1 + x)^n (1 + \frac{1}{x})^n = (1 + x)^n \frac{(x + 1)^n}{x^n} = \frac{(1 + x)^{2n}}{x^n}$.We need the coefficient of $\frac{1

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$\frac{(2n)!}{(n - 1)!(n + 1)!}$

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(B) Given expression: $(1 + x)^n (1 + \frac{1}{x})^n = (1 + x)^n \frac{(x + 1)^n}{x^n} = \frac{(1 + x)^{2n}}{x^n}$.We need the coefficient of $\frac{1}{x}$ in this expansion.This is equivalent to finding the coefficient of $x^{n-1}$ in the expansion of $(1 + x)^{2n}$.The general term in the expansion of $(1 + x)^{2n}$ is given by $T_{r+1} = {}^{2n}C_r x^r$.To find the coefficient of $x^{n-1}$,we set $r = n - 1$.Thus,the coefficient is ${}^{2n}C_{n-1} = \frac{(2n)!}{(n - 1)!(2n - (n - 1))!} = \frac{(2n)!}{(n - 1)!(n + 1)!}$.

The coefficient of $\frac{1}{x}$ in the expansion of $(1 + x)^n (1 + \frac{1}{x})^n$ is

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