Coefficient of , \) in ^{1000}+2 x(1+ \) ^{999}+\mathbf{3} \boldsymbol{x}^{2}(\m

The given sum is S = Σ_{k=0}^{1000} (k+1) x^k (1+x)^{1000-k}. Let y = x/(1+x). Then S = (1+x)^{1000} Σ_{k=0}^{1000} (k+1) y^k = (1+x)^{1000} * (1 - (1002) y^{1001} + (1001) y^{1002})/(1-y)^2. Since 1-y = 1/(1+x), we get S = (1+x)^{1002} - 1002 x^{1001}(1+x) + 1001 x^{1002}. The terms with x^{1001} and x^{1002} are of degree ≥1001, so they do not affect the coefficient of x^50. Hence, the coefficient is the same as that in (1+x)^{1002}, which is C(1002, 50). The option C, though written as \( ^{1002} 250 \), is interpreted as \( \binom{1002}{50} \).