If we have the multiset $$\{1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5\},$$ i.e. five distinct elements and five of each type, how many ways can we partition this multiset into five equally sized parts?
I have looked at some previous posts with similar topic, but they were a bit more general. I was thinking that there may be an easier way to deal with the problem in this specific case.
EDIT: I want the partitions to be ”ordered”, so for example the partition 11111|22222|33333|44444|55555 is counted differently from 22222|11111|33333|44444|55555.
EDIT 2: I would ideally want a way of calculating this by hand, and if possible some sort of generalization.