Nine objects in non-empty boxes

Question

In how many ways 9 identical objects can be put in non-empty boxes of arbitrary size?

Is solution integer partition of 9? That is 30?

Answer 1

We are talking about partitions.

https://en.wikipedia.org/wiki/Partition_(number_theory)

In your case, they are

9

8-1

7-2

7-1-1

6-3

6-2-1

6-1-1-1

5-4

5-3-1

5-2-2

5-2-1-1

5-1-1-1-1

4-4-1

4-3-2

4-3-1-1

4-2-2-1

4-2-1-1-1

4-1-1-1-1-1

3-3-3

3-3-2-1

3-3-1-1-1

3-2-2-2

3-2-2-1-1

3-2-1-1-1-1

3-1-1-1-1-1-1

2-2-2-2-1

2-2-2-1-1-1

2-2-1-1-1-1-1

2-1-1-1-1-1-1-1

1-1-1-1-1-1-1-1-1

Answer 2

Assuming the boxes are distinguishable, we can put up to 8 bars between 9 objects, each bar separating two boxes. There are 2^8 ways to set the bars, 2 meaning a bar is either there or not. So the answer is 256.

Assuming the boxes are indistinguishable, the question becomes sum of number seperation. There are 1 one to use only 1 box, 8 ways to use 2 boxes, 7 ways to use 3 boxes (711,621,531,522,441,432,333), etc.