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.
Best Answer
There are different notions of "ordered" and "unordered" that can arise in relation to set partitions.
all of which are counted differently.
An example of what is counted here is:
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.]
An example of what is counted here is:
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:
which is an ordered set of unordered sets.
To highlight the difference between these two situations, for the first question:
should be considered the same (this is the same as swapping the positions of the first two heaps), whereas in the second question:
should be considered different (this is the same as swapping all the books labelled 1 with all the books labelled 2).