From a standard $52$-card deck, how many ways are there to pick a hand of $k$ cards that includes one card from all four suits?
I know that for any specific $k$, it's possible to break it up into cases based on the partitions of $k$ into $4$ parts. For example, if I want to choose a hand of six cards, I can break it up into two cases based on whether there are $(1)$ three cards from one suit and one card from each of the other three or $(2)$ two cards from each of two suits and one card from each of the other two.
Is there a simpler, more general solution that doesn't require splitting the problem into many different cases?