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Multiset combinations, also known as combinations with repetition, are a concept in combinatorics that deals with selecting elements from a multiset, where the order of selection does not matter and repetitions are allowed. A multiset is a generalization of a set that allows for multiple instances of the same element.

The formula for calculating the number of multiset combinations of n elements taken k at a time is given by the binomial coefficient with repetition, which is expressed as:

C(n + k - 1, k) = (n + k - 1)! / [k! * (n - 1)!]

Here, n represents the number of distinct elements in the multiset, and k is the number of elements to choose. The formula accounts for the possibility of repetition by adjusting the parameters in the binomial coefficient.

For example, consider a multiset with 3 distinct elements {a, b, c} and we want to choose 2 elements with repetition allowed. Using the formula, we get:

C(3 + 2 - 1, 2) = C(4, 2) = 4! / [2! * (4 - 2)!] = 6

This means there are 6 possible combinations: {a, a}, {a, b}, {a, c}, {b, b}, {b, c}, and {c, c}.

Multiset combinations have applications in various fields, including probability theory, statistics, and computer science, where they are used to model scenarios involving repeated selections or events.