Counting sub-multisets of fixed cardinality

TL;DR

This paper derives a formula for counting sub-multisets of fixed size from a multiset, relevant for sampling without replacement from indistinguishable items, a problem not previously published.

Contribution

It introduces a novel expression for the number of sub-multisets of a given size based on element multiplicities, filling a gap in existing literature.

Findings

Provides a new formula for sub-multiset counts

Applicable to sampling without replacement from indistinguishable items

Addresses a previously unpublished problem

Abstract

This report presents an expression for the number of a multiset's sub-multisets of a given cardinality as a function of the multiplicity of its elements. This is also the number of distinct samples of a given size that may be produced by sampling without replacement from a finite population partitioned into subsets, in the case where items belonging to the same subset are considered indistinguishable. Despite the generality of this problem, we have been unable to find this result published elsewhere.