Let's say we have a standard deck of $52$ cards with $13$ cards each of the standard suits of spades, clubs, diamonds, hearts. Let's say we have $p$ players and each player is dealt q cards where $pq<52$.
- Without distinguishing cards by their letter (so we treat any two cards of the same suit as the same, but two cards of different suits as different), how many ways can these cards be distributed?
- How many ways can the cards be dealt so that someone (at least one player) doesn't get a spade?
You can plug in values that you find comfortable for $p$ and $q$, I just need to see work as to how it would be done so I can make it general.
I started working toward this and realized that I can just make the deck a deck of $pq$ cards and then multiply whatever formula I will get by $C(52, pq)$ to cover all of the bases. I don't know where I would go from here.