Distributing $m$ balls into $n$ urns with no urn left empty.

Question

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.

Answer 1

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!

Answer 2

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.