Binomial!

Given expression is the coefficient of $x^4$ in the expansion of :

$(1-(1+x)^{101})^4$

$=\left(\dbinom{101}{1}x+\dbinom{101}{2}x^2 + \cdots \right)^4$

which is $\dbinom{101}{1}^4$

$=(101)^4$

Hence, $\boxed{k=4}$