How to know whether objects can be treated as interchangeable

[frog0101]

Hi,
In questions where they ask you for the number of arrangements of (mn) number of people to be split in to groups of m people (where m and n are positive integers), how do you know whether to divide by m! (I read somewhere that it's based on whether their interchangeable, however, how do you tell??). Such as in this question: In how many ways can 12 different presents be divided into 4 piles of three presents=15400, whereas when the 12 presents are equally divided between 4 children=369 600. Why is this?

Any help would be great 

[Opengangs]

Hey,
So, the way to tell it is purely instinctual. If we divide 12 presents among 4 children, each selection of presents are going to be different because each person is different.

To further demonstrate this idea, let's visualise it.
Suppose we have four people: Adam, Bob, Carter, David.

Now, we have presents labelled [1, 2, ..., 12].
If we choose three presents to give to each, we can do it like this:
Adam: 1, 2, 3
Bob: 4, 5, 6
Carter: 7, 8, 9
David: 10, 11, 12

Or like this:
Adam: 12, 11, 10
Bob: 9, 8, 7
Carter 6, 5, 4
David: 3, 2, 1

We can start to see that these two selections are going to be different. Using that same logic, we can deduce that each selection of (12C3*9C3*6C3) will hold a different set of selections. Now, let's use the same logic to demonstrate why piles need to be divided.

Pile 1: 1, 2, 3
Pile 2: 4, 5, 6
Pile 3: 7, 8, 9
Pile 4: 10, 11, 12

Or:
Pile 5: 12, 11, 10
Pile 6: 9, 8, 7
Pile 7: 6, 5, 4
Pile 8: 3, 2, 1

But wait, Pile 1 is exactly the same as Pile 8. Pile 2 is exactly the same as Pile 7. So, instinctively we know that there will be the overcounting factor of 4!
In short: if we were to divide n objects into r elements, the only time where we divide is if the elements can be counted as the same selection or arrangements, or if they are not distinct.