Probability: Placing books into piles

[TranscendArcu]

Homework Statement

Ten books are made into two piles. In how many ways can this be done if books as well as piles may or may not be distinguishable?

The Attempt at a Solution

So I thought I'd try what I thought was the easiest case first, that being that neither the books nor the piles are distinguishable. In that case, can't the books only be arranged one way? I wouldn't be able to tell the difference between the books in the pile nor the piles themselves, so all combinations would be identical. Is that right?

 

[Dick]

I don't think so. You can still distinguish between the case where there is 1 book in one pile and 9 books in the other pile from the case where there are two books in one pile and 8 books in the other pile.

 

[RoshanBBQ]

Homework Statement

Ten books are made into two piles. In how many ways can this be done if books as well as piles may or may not be distinguishable?

The Attempt at a Solution

So I thought I'd try what I thought was the easiest case first, that being that neither the books nor the piles are distinguishable. In that case, can't the books only be arranged one way? I wouldn't be able to tell the difference between the books in the pile nor the piles themselves, so all combinations would be identical. Is that right?

No. In that case, you still know there is a certain number of books in some pile. You just don't know if it is pile 1 or pile 2.

So if you can distinguish the piles, you can have 6 books in pile 1 or 6 books in pile 2. It cuts the possibilities about in half from the case where you cannot distinguish books but can distinguish piles. Since this one has so few cases, you can just count them:
9 1
8 2
7 3
6 4
5 5So there are 5 possibilities. Note, if you can tell the difference in piles, you would also need to include
4 6
3 7
2 8
1 9

There is also an area of ambiguity in the question. If you put 0 books in a pile, is that still a pile? I'm not sure whether the question wants you to include 10 0.

 

[Ray Vickson]

No. In that case, you still know there is a certain number of books in some pile. You just don't know if it is pile 1 or pile 2.

So if you can distinguish the piles, you can have 6 books in pile 1 or 6 books in pile 2. It cuts the possibilities about in half from the case where you cannot distinguish books but can distinguish piles. Since this one has so few cases, you can just count them:
9 1
8 2
7 3
6 4
5 5


So there are 5 possibilities. Note, if you can tell the difference in piles, you would also need to include
4 6
3 7
2 8
1 9

There is also an area of ambiguity in the question. If you put 0 books in a pile, is that still a pile? I'm not sure whether the question wants you to include 10 0.

Even though you cannot distinguish between the piles, there is still the issue as to how to count the piles. In "classical" probability there are 9 ways of doing it, whether or not you can distinguish between the piles (or rather, whether or not you bother to distinguish between them). In the quantum world there would only be 5 ways, because you could not distinguish between the piles *even if you wanted to*. So, in this problem, do we have not distinguished (but distinguishable-in-principle) piles, or do we have truly absolutely indistinguishable piles? The counts are different, and that makes the probabilities different.

RGV

 

[RoshanBBQ]

Even though you cannot distinguish between the piles, there is still the issue as to how to count the piles. In "classical" probability there are 9 ways of doing it, whether or not you can distinguish between the piles (or rather, whether or not you bother to distinguish between them). In the quantum world there would only be 5 ways, because you could not distinguish between the piles *even if you wanted to*. So, in this problem, do we have not distinguished (but distinguishable-in-principle) piles, or do we have truly absolutely indistinguishable piles? The counts are different, and that makes the probabilities different.

RGV

There are 4 possibilities in terms of being distinguishable:
books are piles are
books are piles aren't
books aren't piles are
books aren't piles aren't

So if in classical probability, there are 9 possibilities when books and piles are not distinguishable, how many possibilities are there when books are not distinguishable but piles are distinguishable?