The coefficient of x^{50} in the expansion of | vedclass

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(D) The given expression is a geometric series with first term $a = (1+x)^{1000}$,common ratio $r = \frac{x}{1+x}$,and number of terms $n = 1001$. Usi

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${}^{1001}C_{50}$

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(D) The given expression is a geometric series with first term $a = (1+x)^{1000}$,common ratio $r = \frac{x}{1+x}$,and number of terms $n = 1001$. Using the sum formula $S_n = \frac{a(1-r^n)}{1-r}$: $f(x) = \frac{(1+x)^{1000} \left(1 - (\frac{x}{1+x})^{1001}\right)}{1 - \frac{x}{1+x}}$ $f(x) = \frac{(1+x)^{1000} - \frac{x^{1001}}{1+x}}{\frac{1+x-x}{1+x}} = \frac{(1+x)^{1001} - x^{1001}}{1} = (1+x)^{1001} - x^{1001}$. The coefficient of $x^{50}$ in $(1+x)^{1001} - x^{1001}$ is the coefficient of $x^{50}$ in $(1+x)^{1001}$,which is ${}^{1001}C_{50}$.

The coefficient of $x^{50}$ in the expansion of $(1+x)^{1000} + x(1+x)^{999} + x^2(1+x)^{998} + \ldots + x^{1000}$ is

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