Understanding the Multinomial Coefficient and its Independence in Combinatorics

[aaaa202]

Okay I got to wonder about this: Why is the multinomial coefficient independent of if you start by taking out n1, n2, n3 etc. or n2,n1,n3 or n3, n2, n1... etc.. I mean intuitively from actually doing the combinatorics by counting it seems obvious that the order should not matter. But can this be proved or is it taken as an axiom?

 

[Vargo]

Because no matter how you word it, it is the number of ways to partition a set of size N into subsets with respective sizes n1, n2, ..., nk. And there is no "ordering" of the subsets, so you interpret it in any order you like.