# Dealing with grouping problem in combinatorics

I am trying to solve some problems based on this formula,but am facing some issues in determine whether or not consider ordering as important.

For Example:

In how many ways 15 different books can be divided five heaps of equal number of books ?

So here according to the answer the ordering of groups is not considered important,hence the answer given is $\frac {15!} {5!(3!)^5}$

and the same thing goes for this one :

In how many ways 3 piles can be formed out of 18 different books, which will be $\frac {18!} {3!(6!)^3}$

But for the following problem "There are three copies, each of four different books. In how many ways they can be arranged on a self ?"

This this is similar to divide 12 books into 4 sets of 3 books each, the answer given here considers the ordering important,why this change ?

Why in the first two the ordering not and on the third one it is important ? I guess (if the solution is correct) it is something related to copies but am not confident enough.

A good example of a problem where ordering should be considered important is say "In how many ways can 18 different books be divided equally among three students ?"

The solution should be $\frac {18!} {(6!)^3}$

But I am not much confident while in these problems so I will be grateful if somebody helps me to understand when to consider ordering important and when not to.

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There are different notions of "ordered" and "unordered" that can arise in relation to set partitions.

• An unordered set of unordered sets,
• An unordered set of ordered sets,
• An ordered set of unordered sets,
• An ordered set of ordered sets,

all of which are counted differently.

In how many ways 15 different books can be divided five heaps of equal number of books ?

An example of what is counted here is:

{{1,3,8},{2,14,15},{4,5,7},{6,12,13},{9,10,11}}


is an unordered set of unordered sets. This is what the answer to the first question suggests it's counting. [Although, the question itself does not resolve the question of order within the heaps.]

There are three copies, each of four different books. In how many ways they can be arranged on a self ?

An example of what is counted here is:

1 2 3 1 4 1 3 3 2 2 4 4


which we can interpret as a set partition counting problem -- place the i-th element in the s(i)-th set, where s(i) denotes the element in the i-th position. So the above example gives rise to:

({1,4,6},{2,9,10},{3,7,8},{5,11,12})


which is an ordered set of unordered sets.

To highlight the difference between these two situations, for the first question:

{{1,3,8},{2,14,15},{4,5,7},{6,12,13},{9,10,11}}
{{2,14,15},{1,3,8},{4,5,7},{6,12,13},{9,10,11}}


should be considered the same (this is the same as swapping the positions of the first two heaps), whereas in the second question:

({1,4,6},{2,9,10},{3,7,8},{5,11,12})
({2,9,10},{1,4,6},{3,7,8},{5,11,12})


should be considered different (this is the same as swapping all the books labelled 1 with all the books labelled 2).

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