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 May21 awarded Scholar May21 accepted Distribute stones into buckets with limits May21 comment Distribute stones into buckets with limits Thanks, you answered my question. I totally forgot about generating functions (studied this stuff 15 years ago) and that one works. May20 comment Distribute stones into buckets with limits Thanks for your answer too. This seems to assume the stones are indistinguishable. However, as I mentioned to svenkatr, in my question the stones are all different from each other. Sorry if it was unclear -- I have since updated the question. A side question, though: How do you use the Binomial Theorem to expand $(1 - x)^{-b}$? May20 awarded Editor May20 revised Distribute stones into buckets with limits Clarified the problem and added examples. May20 comment Distribute stones into buckets with limits Thanks for your answer. I probably should have been more clear; when I wrote that the stones were "different", I meant to say that they are distinguishable. In that case, there are actually 6 ways to distribute 4 stones in 2 buckets, where bucket holds at most 2 stones: (AB, CD), (AC, BD), (AD, BC), (BC, AD), (BD, AC) and (CD, AB). If you have 3 buckets instead of 2, I think the answer is 54, according to the Python script. May20 awarded Student May20 asked Distribute stones into buckets with limits May20 awarded Autobiographer