Stars and bars

Builds on Binomial coefficient Combinations with repetition

Statement

Let $n$ and $k$ be natural numbers with $n \ge 1$ . Then the number of multisets of size $k$ with elements in an $n$ -element set is $\left(\!\binom{n}{k}\!\right) = \binom{n + k - 1}{k}.$ Equivalently, this is the number of solutions $(x_1, \dots, x_n)$ of $x_1 + \dots + x_n = k$ in natural numbers.

Remarks

The proof writes a multiset as a row of $k$ stars and $n - 1$ bars. The bars separate the stars into $n$ groups, one group for each element. The number of solutions in positive integers is $\binom{k - 1}{n - 1}$ for $k \ge 1$ . This is the number of compositions of $k$ into $n$ parts.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Tag	`009TEMTQKH`
URI	`m2://009TEMTQKH`
Kind	statement, theorem
Writing	W0 stub
MSC	05A10
Status	This copy of the site has no build data. Run `m2 status 009TEMTQKH`.
Uses	Binomial coefficient, Combinations with repetition
Same as	Wikidata Q7602343 Wikipedia Stars and bars (combinatorics)

Stars and bars

Statement

Remarks

Reading path

Further reading

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