If $m \geq n$, how many different ways are there of distributing $m$ indistinguishable balls into $n$ distinguishable urns with no urn left empty? I have no idea how to even start with this so any help would be greatly appreciated. Thanks in advance.
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2This is a standard Stars and Bars problem. Go to the Wikipedia article, it is quite thorough. The answer will be $\binom{m-1}{n-1}$. – André Nicolas May 14 '13 at 02:53
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See also this highly related answer. – Cameron Buie May 14 '13 at 03:27
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HINT: To fulfill the "no urn left empty" clause, you must distribute $n$ balls into $n$ urns, one in each.
You can then distribute the remaining $(m-n)$ balls into $n$ urns however you like.
xisk
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Are you familiar with how to calculate the number of distributions of $m$ indistinguishable balls to $n$ distinguishable urns without any restrictions? If so, here's a hint:
Hint: Distribute 1 ball to each of the $n$ urns before doing anything. Then the number of distributions of $m$ indistinguishable balls to $n$ distinguishable urns with no urn left empty is the same as the number of distributions of $m-n$ indistinguishable balls to $n$ distinguishable urns with no restrictions.
If you need more help on the second calculation, feel free to post and I'll post more information!
Alex Wertheim
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