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The coefficient of $x^n$ in the expansion $(1+x)(1-x)^n$ is

$\begin{array}{1 1} =(-1)^n [1-n] \\ =(-1)^n [1+n] \\ =(-1)^{n-1} n \\ =(-1)^n [n-1] \end{array}$

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Coefficient of $x^n$ in $(1+x)(1-x)^n$$= 1\times$ coeff. of $ x^n $ in $(1-x)^n+1\times$ coeff. of $x^{n-1}$ in $(1-x)^n$
$\therefore$ Coefficient of $x^n$ in $ (1+x)(1-x)^n$$=(-1)^n $$ ^nC_n $$+(-1)^{n-1}$$ ^nC_{n-1}$
$=(-1)^n [1-n]$
answered Sep 15, 2013 by rvidyagovindarajan_1

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