Just (not) Binomial Theorem

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

If the coefficient of \(x^{50}\) in the above expression is in the form \(\dbinom{1002}{k}\) and \(k<100\), then find the positive integer \(k\).

\[\] Notation: \( \dbinom MN = \dfrac {M!}{N! (M-N)!}\) denotes the binomial coefficient.

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