 TOPICS  # Multichoose

The number of multisets of length on symbols is sometimes termed " multichoose ," denoted by analogy with the binomial coefficient . multichoose is given by the simple formula where is a multinomial coefficient. For example, 3 multichoose 2 is given by 6, since the possible multisets of length 2 on three elements are , , , , , and .

The first few values of are given in the following table. 1 2 3 4 5 1 1 2 3 4 5 2 1 3 6 10 15 3 1 4 10 20 35 4 1 5 15 35 70 5 1 6 21 56 126

Multichoose problems are sometimes called "bars and stars" problems. For example, suppose a recipe called for 5 pinches of spice, out of 9 spices. Each possibility is an arrangement of 5 spices (stars) and dividers between categories (bars), where the notation indicates a choice of spices 1, 1, 5, 6, and 9 (Feller 1968, p. 36). The number of possibilities in this case is then ,

Ball Picking, Binomial Coefficient, Choose, Combination, Figurate Number, Hypergeometric Distribution, Multinomial Coefficient, Multiset, Permutation, String

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## References

Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed. New York: Wiley, 1968.Scheinerman, E. R. Mathematics: A Discrete Introduction. Pacific Grove, CA: Brooks/Cole, 2000.

Multichoose

## Cite this as:

Weisstein, Eric W. "Multichoose." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Multichoose.html