But, it's important to appreciate that, even for probabilities on finite or countable sets, you get pulled into continuous math. I don't think we disagree here, but maybe some examples would be worthwhile, just for definiteness.
Expectations (means, variances) of discrete variables will be real-valued. There are continuous processes that are intimately connected to what you thought were purely discrete outcomes (e.g., the relationship between Poisson counts and exponential waiting times). There will be limiting processes of discrete structures that bring in continuous probabilities (the Binomial -> normal limit, and all its generalizations) and provide considerable insight. The Stirling formula, which is key to insight about factorials, comes from calculus.
Finally, generating functions are one of the main tools for solving the discrete summations you mention, and these are continuous, and pull you straight into complex analysis. ("Who changed the subject here? I was just trying to add some binomial coefficients and now we're talking about derivatives of analytic functions at zero?")
Or, as my edition of Concrete Mathematics says (sec 5.4): "We come now to the most important idea in this whole book, the notion of a generating function."
Probability does use some tools from combinatorics and vice versa, but the two subjects have vastly different goals.
Similarly I find it highly questionable whether combinatorics is "clearly more relevant to general programming". In fact, what is "general programming"? Probability theory pervades all of Computer Science - from probabilistic algorithms (QuickSort), through cryptography, optimization algorithms (genetic algorithms, simulated annealing), networking (information theory, queuing theory), machine learning, the list goes on and on. For business programming, statistics (based on probablity theory) is crucial. Some of those applications include combinatorics, but I find it harder to find such a long list of applications of combinatorics being used without probability theory - it is a more specialized field.