Stars and bars

Stars and bars is a counting method that represents identical objects as stars and group boundaries as bars. It counts distributions among distinct groups by arranging these symbols.

Combinations: A way to count selections in which order does not matter. Choosing bar positions is a combination because the separators are indistinguishable.
Occupancy problem: A counting or probability problem about placing objects into containers. Stars and bars solves occupancy counts when objects are identical and containers distinct.
Combinatorics: The branch of mathematics concerned with counting, arrangement, and selection of discrete objects. Stars and bars is a basic technique for counting discrete allocations.
Stirling number of the second kind: The number of ways to partition a set of labeled elements into a fixed number of nonempty unlabeled subsets. It counts partitions of distinct objects, unlike distributions of identical objects.
Binomial coefficient: The number of ways to choose a fixed-size subset from a finite set. It gives the formula for choosing separator positions among stars and bars.
Multinomial distribution: A probability distribution for counts of outcomes across several categories in repeated trials. Its outcome-count coefficients count ways to allocate labeled trials among categories.
Permutation: An ordered arrangement of distinct objects. Stars and bars encodes allocations without treating identical objects as distinct.
Labeled balls into labeled boxes: An allocation model in which both objects and containers are individually distinguishable. Distinguishable objects require choices beyond the separator arrangements.
Weak composition: An ordered list of nonnegative integers whose sum is a specified total. Each stars-and-bars arrangement encodes one such distribution.
Binomial distribution: A probability distribution for the number of successes in a fixed number of independent trials. It is the two-category case, while stars and bars counts allocations of identical objects.

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