From a standard 52-card deck, how many ways are there to pick a hand of...
From a standard -card deck, how many ways are there to pick a hand of cards that includes one card from all four suits?
I know that for any specific , it's possible to break it up into cases based on the partitions of into 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 three cards from one suit and one card from each of the other three or 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?