Find the coefficients of `x^(50)` in the expression `(1+x)^(1000)+2x(1+x)^(999)+3x^2(1+x)^(998)+....+1001 x^(1000)` .

Explanation

To find the coefficient of \( x^{50} \) in the expression

\[ (1+x)^{1000} + 2x(1+x)^{999} + 3x^2(1+x)^{998} + \ldots + 1001 x^{1000}, \]

we can denote the entire expression as \( S \).

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