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Problem: Let's say we have set of "N = 12" objects of which 3 are identical footballs, 4 are identical tennis balls and 5 are identical golf balls.

Let's say we have 5 buckets, of which two are Blue, and one each is Red, Green and Yellow.

How many ways are there to arrange these 12 objects into the five buckets specified above? There are no limitations on how the objects are arranged: any bucket can get all (12) or none (0) of the objects, or any number in between.

A solution is needed for the case where all of the objects must go into the buckets (none are left outside), and when any number of the objects from 0..12 can be left outside.

Comments: I'm looking for a methodical step-by-step approach to solving the problem specified above. I'm trying to learn about permutations and combinations, and I've made up the problem with features that I find confusing. I'm hoping that seeing a solution to this problem will help me learn how to go about distributing a set of identical and non-identical objects into identical and non-identical buckets. I've pretty much thrown the kitchen sink into this problem as you can see.

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  • $\begingroup$ We could solve the problem for these particular small numbers. However, indistinguishable buckets add an unpleasant extra layer of complexity, because (in general) we end up looking at partitions, where we tend not to have nice closed forms. $\endgroup$ – André Nicolas Aug 10 '15 at 5:24
  • $\begingroup$ Thanks for your comment, @AndréNicolas. Could you please get me started on solving this problem without the indistinguishable buckets? Again, I only know the very basis of permutations and combinations, so a general solution is anyways beyond my comprehension at the moment. $\endgroup$ – user3837690 Aug 10 '15 at 6:54
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First thing, carefully study Theorem 2 of the "stars and bars" formula, fully explained here

Then apply the formula separately for each type of item, and multiply, thus

$${3+5-1\choose 5-1}{4+5-1\choose 5-1}{5+5-1\choose 5-1} $$

$$ = {7\choose 4}{8\choose 4}{9\choose 4}$$


I misread the question as distributing distinct objects to identical boxes. No harm done, you now know how to tackle such cases.

It is not a good idea to throw in the kitchen sink, as you express it: far better to first systematically study more elementary cases. There are quite a lot of other types, and rather than try to give lengthy explanations here, you can see "Counting objects in boxes", and study the types that interest you.

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