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

Question

$(a)\;\large\frac{(1000)!}{(50)!(950)!}\qquad(b)\;\large\frac{(1000)!}{(49)!(951)!}\qquad(c)\;\large\frac{(1001)!}{(51)!(950)!}\qquad(d)\;\large\frac{(1001)!}{(50)!(951)!}$