I'll start with a specific example of what I am trying to solve:
I have eight balls to be randomly placed into four buckets. Buckets #1-3 have the capacity of 2, 2, 3 respectively, while bucket #4 has an infinite capacity. A bucket can't be filled over its capacity. A ball will not be thrown to a bucket that is already full. I would like to calculate the probability that my eight balls will completely fill buckets #1-3.
The last time I studied maths/stats formally was in university, a good 7-10 years ago. My limited memory/understanding of combinatorics+probability is failing me. I'm a software guy, so I wrote a basic simulator for the problem, which seems to tell me that my eight balls will fill buckets #1-3 ~18% of the time. I'd like to understand how to approach the problem mathematically.