I was working on this problem:
Let N be a geometric random variable with parameter p. Suppose that the conditional distribution of X given that N = n is the gamma distribution with parameters n and λ. Find the conditional probability mass function of N given that X = x.
And in the solution, they started with $$P(N=n|X=x)=\frac{f_{X|N}(x|n)P(N=n)}{f_{X}(x)}$$
However why is that they can simply combine the p.d.f. of X conditioned on N with the p.m.f. of N? Pmf expresses probability while pdf is a probability density.
Bayes' Theorem does hold with $N$ being a discrete r.v. and $X$ being a continuous r.v. You can see page 6 of this document for a proof.