Understanding the Multinomial Coefficient and its Independence in Combinatorics

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The multinomial coefficient is independent of the order in which subsets are selected because it represents the number of ways to partition a set of size N into subsets of specified sizes n1, n2, ..., nk. This independence is rooted in the combinatorial interpretation that the arrangement of subsets does not affect the total count of partitions. While intuitive, this property can be formally proven through combinatorial arguments or algebraic identities. The lack of ordering among subsets reinforces that the multinomial coefficient remains constant regardless of the sequence of selection. Thus, it is a fundamental aspect of combinatorial mathematics.
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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?
 
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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.
 

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