Explain the mistake in the following “counting argument”: Suppose we want to count the number of ways that r indistinguishable balls can be placed in n distinct bins, under the assumption that each bin can take more than one ball and that the order of the balls inside the bin does not count. Assume first that the balls are distinct. Then there are $n^r$ ways to place them into the bins under the same assumptions. Divide this number by $r!$ to make up for the fact that the balls are indistinguishable to get $n^r/ r! $.
2026-03-26 21:26:25.1774560385
Balls and bins confusion, why can't we just divide by r!
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What if all the balls are in one bin? Which $r!$ different configurations are you dividing out then?