The coefficient of x^n in the expansion of (1 | vedclass

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(B) The expression is $(1 + x)(1 - x)^n = (1 - x)^n + x(1 - x)^n$. We need to find the coefficient of $x^n$ in this expression. The coefficient of $x^

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$(-1)^n(1 - n)$

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(B) The expression is $(1 + x)(1 - x)^n = (1 - x)^n + x(1 - x)^n$. We need to find the coefficient of $x^n$ in this expression. The coefficient of $x^n$ in $(1 - x)^n$ is given by the general term $T_{r+1} = \binom{n}{r} (1)^{n-r} (-x)^r = \binom{n}{r} (-1)^r x^r$. For $r = n$,the coefficient is $\binom{n}{n} (-1)^n = (-1)^n$. The coefficient of $x^n$ in $x(1 - x)^n$ is the coefficient of $x^{n-1}$ in $(1 - x)^n$. For $r = n-1$,the coefficient is $\binom{n}{n-1} (-1)^{n-1} = n (-1)^{n-1} = -n (-1)^n$. Adding these,the total coefficient is $(-1)^n - n(-1)^n = (-1)^n(1 - n)$.

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

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