I am into a CCG, and I got a question come to mind "how many possible out comes are there for deck combinations?"
The game is broken into three: Main Character (6 cards available, only one deck), Draw Deck (176 cards available, up to three copies, 45-528 card deck), Problem Deck (35 cards available, up to 2 copies, 10-70 card deck).
I wan't to know how many combinations for each (such as 6 for main characters and 6 for all three parts at max, take one from the draw deck[527] then you have 529 for just the draw deck and 3,174 for all three parts), and don't care about the order they come in. I would like to take the formula to also find more likely combinations (i.e. playable) if at all possible.
You can consider each deck separately and multiply the number of options for each deck. For the Draw Deck, each card can come in zero to three copies, so without the $45$ card minimum you would have $176^4=959512576$ possibilities. The practical answer is that the choices less than $45$ cards will be a small fraction of this, so ignore it. Similarly for the Problem Deck, without the $10$ card minimum you would have $35^3=42875$ choices, giving about $2.5\cdot 10^{14}$ choices.