I see three reasons.
Generating functions is an example of tools used in discrete mathematics. Calculus definitely helps working with them.
Binomial coefficients arise frequently in discrete math. Many formulas about these coefficients can be handled by calculus.
Also, even if you are interested only on what happens for finite sets of size n, probably you will want to let n goes to infinity at some point, and then continuous laws, integrals and the like will appear naturally.
Still I think that it is possible to teach a beginner course in discrete mathematics which does not rely on calculus.