In how many ways can we place $n$ indistinguishable balls in $n$ urns so that exactly one urn is empty?

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How many ways can we place $n$ indistinguishable balls in $n$ urns so that exactly one urn is empty?

So if I do this similar to stars and bars I am looking to put $n$ balls in actually $n-1$ urns, so $I$ thought let me do that. So place one ball in each of $n-1$ urns, which should leave me with $1$ ball. Now since I need one urn empty, that would occur in $n$ ways so there are $n$ ways to do this.

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2 Answers

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Choose the empty urn and choose the urn containing two balls. Ans: n(n-1)

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Or to imagine it a little differently, you put one ball in each cup, now you remove a ball from one cup and put it in any other, giving you one empty cup and one with two balls. When picking the cup from which you take the ball you had n options and when you put the ball back you had n-1 options since you could not return the ball in the same cup, hence n*(n-1)

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