[Math] Ways to put $n$ balls into $m$ boxes

Question

There are several different versions of the problem:

1. The balls are distinct. The boxes are distinct.

2. The balls are indistinct. The boxes are distinct.

3. The balls are distinct. The boxes are indistinct.

4. The balls are indistinct. The boxes are indistinct.

I think the answer for 1, 2 are easy. Just $m^n$ and $\tbinom{n+m-1}{n}$.

However, for 3, 4, I really have no idea. For 3, If $\frac{m}{n}\gg1$, I think the answer is approximate $\frac{m^n}{m!}$.

Answer 1

These are 4 of the 12 problems in "the twelvefold way".