What is the exact probability of rolling every possible result on an a-sided die at least b times each after rolling c times?

Question

I have found answers to similar, much simpler questions on this site before (such as here How do you calculate probability of rolling all faces of a die after n number of rolls?), but can't find anything that deals with this exact generalization. After thinking about it some, this seems headache-inducingly more complicated to solve than these simpler variants, because the combinatorics are more nested and interfere with the simpler approaches used to solve those simpler problems (e.g. the fact that you want to roll each number b times instead of just once as in the linked question means that you can't use such a simple expression for not rolling that number in c rolls).