Dividing some balls into some boxes

Question

As we know, if distinguishable balls and indistinguishable boxes, the answer is Stirling's Number of the second type. If indistinguishable balls and distinguishable boxes, we use Euler's candy division.

What about indistinguishable balls and indistinguishable boxes?

Answer 1

Let me give you an example ,

How many ways are there to pack $8$ identical DVD into 5 identical boxes so that each box contains at least one DVD

Now ,we should use patition method such that $4-1-1-1-1$ or $3-2-1-1-1$ or $2-2-2-1-1$

Some hints for partition:

1-) Generating functions can be used for partitions.

2-) Moreover, $p(k) \approx \frac{e^{\pi \sqrt{2k/3}}}{4k\sqrt{3}}$ can be used for large number of partition

How many ways are there to pack $9$ identical DVD into 3 identical boxes so that each box contains at least two DVD

Answer: $5-2-2$ or $4-3-2$ or $3-3-3$