Areteem: A Random Logic Question on Probability

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A Random Logic Question on Probability

A permutation is used to put unique objects into unique boxes. Combinations put unique objects into identical boxes. Stars and bars put identical objects into unique boxes. Then how do you find putting identical objects into identical boxes? I really don't know a method besides counting it out.

well if they are all identical then there is only 1 way, is there not?

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Well say I want to add three positive intergers to a total of 7. That means that I get (1,2,4), (1,1,5), (1,3,3), (2,2,3), as my only choices, totaling 4. This is not stars and bars, as stars and bars would do order. This is just having an identical set of 7 ones and figuring out how to split it into ununique groups.

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There is no way to count it unless you count it one-by-one, using casework. The only case in which double-indistinct is avoidable is when you have 2 boxes. Then it is (S+B)/2!

However, this does not generalize to n boxes. If you have 3 boxes, you cannot do this because some cases cannot be permutated 3!=6 ways.

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The general case for this problem is pretty hard.

However, you should be able to derive a formula for 3 boxes.

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