# Multinomial theorem

The multinomial theorem is a formula in algebra and combinatorics that expands a power of a sum of any number of terms, expressing \( (x_1 + x_2 + \cdots + x_k)^n \) for a nonnegative integer \( n \) as a sum of monomials whose coefficients are the multinomial coefficients \( n!/(n_1! n_2! \cdots n_k!) \).<sup>[1](https://dlmf.nist.gov/26.4)</sup> It generalizes the binomial theorem from two terms to \( k \) terms, and it underlies the multinomial distribution in probability theory.<sup>[2](https://mis.alagappauniversity.ac.in/siteAdmin/dde-admin/uploads/6/__UG_B.Sc._Mathematics_113%2064_Combinatorics_BSc%20%28Mathematics%29_9494.pdf)</sup><sup> • </sup><sup>[3](https://faculty.washington.edu/yenchic/20A_stat512/Lec7_Multinomial.html)</sup>

| Key fact | Detail |
|---|---|
| Statement | \( (x_1 + \cdots + x_k)^n = \sum_{n_1 + \cdots + n_k = n} \binom{n}{n_1, \ldots, n_k} x_1^{n_1} \cdots x_k^{n_k} \), summed over nonnegative integers \( n_i \)<sup>[1](https://dlmf.nist.gov/26.4)</sup> |
| Coefficient | \( \binom{n}{n_1, \ldots, n_k} = \frac{n!}{n_1! n_2! \cdots n_k!} \), the number of ways of placing \( n \) distinct objects into \( k \) labeled boxes with \( n_j \) objects in the \( j \)th box<sup>[1](https://dlmf.nist.gov/26.4)</sup> |
| Number of terms | The expansion of \( (x_1 + \cdots + x_k)^n \) contains \( \binom{k - 1 + n}{n} \) distinct terms<sup>[4](https://ww2.coastal.edu/dweathers/edm/multinomials.html)</sup> |
| Sum of coefficients | Setting every \( x_i = 1 \) gives \( \sum \binom{n}{n_1, \ldots, n_k} = k^n \); for three terms the sum is \( 3^n \)<sup>[5](https://math.uchicago.edu/~alephnil/notes/MATH-465-notes.pdf)</sup> |
| Standard proof | Induction on the number of variables \( m \), using the binomial theorem in the inductive step<sup>[5](https://math.uchicago.edu/~alephnil/notes/MATH-465-notes.pdf)</sup> |
| History | Traced in German scholarship to correspondence between Leibniz and Johann Bernoulli in 1695; Abraham de Moivre proved it in 1697<sup>[6](https://veridicaldatasciencebook.stat.berkeley.edu/~census/neylogit.pdf)</sup><sup> • </sup><sup>[7](https://mathshistory.st-andrews.ac.uk/SH/de_moivre_sh.pdf)</sup> |
| Probabilistic role | The multinomial distribution's probabilities \( \frac{n!}{x_1! \cdots x_k!} p_1^{x_1} \cdots p_k^{x_k} \) sum to 1 precisely because of the multinomial expansion<sup>[3](https://faculty.washington.edu/yenchic/20A_stat512/Lec7_Multinomial.html)</sup> |

## Statement and coefficients

For a nonnegative integer \( n \) and variables \( x_1, x_2, \ldots, x_k \), the theorem states<sup>[1](https://dlmf.nist.gov/26.4)</sup>

\[ (x_1 + x_2 + \cdots + x_k)^n = \sum_{n_1 + n_2 + \cdots + n_k = n} \binom{n}{n_1, n_2, \ldots, n_k} x_1^{n_1} x_2^{n_2} \cdots x_k^{n_k}, \]

where the sum runs over all nonnegative integers \( n_1, n_2, \ldots, n_k \) adding to \( n \). The coefficient is the multinomial coefficient

\[ \binom{n}{n_1, n_2, \ldots, n_k} = \frac{n!}{n_1! n_2! \cdots n_k!}. \]

It also factors as a product of binomial coefficients,<sup>[1](https://dlmf.nist.gov/26.4)</sup>

\[ \binom{n}{n_1, \ldots, n_k} = \prod_{j=1}^{k-1} \binom{n_j + n_{j+1} + \cdots + n_k}{n_j}, \]

and for two terms it reduces to the ordinary binomial coefficient \( \binom{n_1 + n_2}{n_1} \).<sup>[1](https://dlmf.nist.gov/26.4)</sup> The coefficient counts the distinct permutations of a multiset of \( k \) distinct elements with multiplicities \( n_i \), equivalently the \( k \)-dimensional lattice paths from the origin to \( (n_1, \ldots, n_k) \).<sup>[8](https://mathworld.wolfram.com/MultinomialCoefficient.html)</sup><sup> • </sup><sup>[1](https://dlmf.nist.gov/26.4)</sup>

A worked example: in the expansion of \( (x + y + z)^7 \), the coefficient of \( x^2 y^2 z^3 \) is \( 7!/(2!\,2!\,3!) = 210 \), the coefficient of \( x y z^5 \) is \( 7!/(1!\,1!\,5!) = 42 \), and the coefficient of \( x^3 z^4 \) is \( 7!/(3!\,0!\,4!) = 35 \).<sup>[9](https://www.ms.uky.edu/~dmu228/ma320/lectures/sec3.2.pdf)</sup>

**Counting terms.** Because the expansion collects like monomials, the number of distinct terms equals the number of multisets of cardinality \( n \) drawn from \( k \) elements, which is \( \binom{k - 1 + n}{n} \).<sup>[4](https://ww2.coastal.edu/dweathers/edm/multinomials.html)</sup> For example, \( (a_1 + a_2 + a_3)^3 \) has ten terms, including the mixed term \( 6 a_1 a_2 a_3 \).<sup>[10](https://mathworld.wolfram.com/MultinomialSeries.html)</sup>

## Proofs

The standard proof proceeds by induction on the number of variables, applying the binomial theorem at the inductive step.<sup>[5](https://math.uchicago.edu/~alephnil/notes/MATH-465-notes.pdf)</sup>

**A probabilistic proof.** K. K. Kataria, a mathematician who published the argument in 2016, derived the theorem from the multinomial distribution: since the joint probability mass function \( n! \prod_{j=1}^{m} p_j^{k_j} / k_j! \) is a valid distribution, its probabilities sum to 1, and substituting \( p_i = x_i / (x_1 + \cdots + x_m) \) recovers the expansion for positive reals, with the general case following by polynomial identity.<sup>[11](https://arxiv.org/abs/1608.01718)</sup> This proof avoids both mathematical induction and the binomial theorem.<sup>[11](https://arxiv.org/abs/1608.01718)</sup>

The theorem holds over any commutative ring or field, not only over the reals.<sup>[12](https://amiso.my/assets/files/ysc/Theorems_and_lemmas_in_mathematics.pdf)</sup>

## History

The multinomial theorem grew out of the binomial theorem, which [Isaac Newton](https://www.edgepedia.org/isaac-newton) had generalized to fractional and negative powers as an infinite series.<sup>[13](https://nzdr.ru/data/media/biblio/kolxoz/M/MGe/Greenwald%20S.J.%2C%20Thomley%20J.E.%20Encyclopedia%20of%20mathematics%20and%20society.%20Vol.1-3%20%28Salem%2C%202012%29%28O%29%28ISBN%209781587658440%29%281252s%29_MGe_.pdf)</sup> German scholarship traces the multinomial case, the "polynomische Lehrsatz", to correspondence between Leibniz and [Johann Bernoulli](https://www.edgepedia.org/johann-bernoulli) in 1695.<sup>[6](https://veridicaldatasciencebook.stat.berkeley.edu/~census/neylogit.pdf)</sup> MathWorld credits the multinomial series, the generalization of the [binomial series](https://www.edgepedia.org/binomial-series), to Johann Bernoulli and Leibniz.<sup>[10](https://mathworld.wolfram.com/MultinomialSeries.html)</sup>

**De Moivre's proof.** Abraham de Moivre (1667–1754), the French-born mathematician who settled in London, generalized Newton's binomial theorem to the multinomial theorem; the Royal Society learned of the method in 1697 and elected him a member that year.<sup>[7](https://mathshistory.st-andrews.ac.uk/SH/de_moivre_sh.pdf)</sup> His 1697 paper in the *Philosophical Transactions* was titled "A method of raising an infinite multinomial to any given power, or extracting any given root of the same", and it established a law for expanding a given algebraic expression, even an infinite series, to any positive integer power.<sup>[14](https://www.cambridge.org/core/books/abs/series-and-products-in-the-development-of-mathematics/bibliography/A4F426D10F2E22AAA62ADA73C0248CB3)</sup><sup> • </sup><sup>[15](https://www.tandfonline.com/doi/full/10.1080/26375451.2026.2648415)</sup> In Britain, this proof served as an important preliminary for later eighteenth-century attempts to prove the binomial theorem itself without fluxional methods.<sup>[15](https://www.tandfonline.com/doi/full/10.1080/26375451.2026.2648415)</sup> The 1911 *Encyclopædia Britannica* likewise described the multi- (or poly-) nomial theorem as the problem of expanding any power of a multinomial, discussed in 1697 by Abraham de Moivre.<sup>[16](https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Binomial)</sup>

## Applications in probability

The theorem guarantees the normalization of the multinomial distribution, the generalization of the binomial distribution to \( k \) categories.<sup>[3](https://faculty.washington.edu/yenchic/20A_stat512/Lec7_Multinomial.html)</sup> If \( X \sim M_k(n; p_1, \ldots, p_k) \) counts the outcomes of \( n \) independent trials falling into \( k \) mutually exclusive categories with probabilities \( p_1, \ldots, p_k \), then

\[ P(X_1 = x_1, \ldots, X_k = x_k) = \frac{n!}{x_1! x_2! \cdots x_k!} p_1^{x_1} \cdots p_k^{x_k}, \]

and summing over all \( x_i \) with \( \sum_i x_i = n \) gives 1 exactly by the multinomial expansion with \( a_i = p_i \).<sup>[3](https://faculty.washington.edu/yenchic/20A_stat512/Lec7_Multinomial.html)</sup> The multinomial coefficient itself is the number of ways to put \( n \) balls into \( k \) boxes.<sup>[3](https://faculty.washington.edu/yenchic/20A_stat512/Lec7_Multinomial.html)</sup>

The distribution inherits structure from the theorem's coefficient algebra. Each marginal count \( X_j \) follows a binomial distribution \( \mathrm{Bin}(n, p_j) \), and any merged count \( X_i + X_j \) follows \( \mathrm{Bin}(n, p_i + p_j) \); merging categories of a multinomial vector yields a new multinomial distribution with the combined probabilities.<sup>[3](https://faculty.washington.edu/yenchic/20A_stat512/Lec7_Multinomial.html)</sup> In statistical mechanics and combinatorics, multinomial coefficients arise naturally from binomial coefficients when a number distribution of labels is placed on a set of items.<sup>[2](https://mis.alagappauniversity.ac.in/siteAdmin/dde-admin/uploads/6/__UG_B.Sc._Mathematics_113%2064_Combinatorics_BSc%20%28Mathematics%29_9494.pdf)</sup>

## Extensions

The theorem has been extended in several directions. A \( q \)-multinomial coefficient, defined in the NIST Digital Library of Mathematical Functions, counts multiset permutations weighted by inversion number or major index.<sup>[17](https://dlmf.nist.gov/26.16)</sup> Noncommutative versions exist: a free noncommutative binomial (or multinomial) theorem has been formulated in terms of the Lyndon-Shirshov basis, and the Bell differential polynomials of Schimming and Rida give a binomial theorem for any associative algebra, extendable to the multinomial case.<sup>[18](https://arxiv.org/abs/2304.06432)</sup> Marinus Schlosser has proved an elliptic extension of the multinomial theorem for elliptic commuting variables, extending his earlier elliptic binomial theorem to higher rank.<sup>[19](https://arxiv.org/abs/2307.12921)</sup> Computational work has also addressed the practical side: factorial terms in multinomial coefficients grow rapidly, motivating efficient methods for determining specific coefficients in multinomial expansions.<sup>[20](https://www.researchgate.net/publication/402709586_Efficient_Methods_for_Determining_Coefficients_in_Multinomial_Expansions)</sup>

## References

1. [DLMF §26.4 Lattice Paths: Multinomial Coefficients and Set Partitions, NIST Digital Library of Mathematical Functions](https://dlmf.nist.gov/26.4)
2. [UNIT 8 Multinomials, Alagappa University Combinatorics course material](https://mis.alagappauniversity.ac.in/siteAdmin/dde-admin/uploads/6/__UG_B.Sc._Mathematics_113%2064_Combinatorics_BSc%20%28Mathematics%29_9494.pdf)
3. [UW STAT 512, Lecture 7: Multinomial distribution](https://faculty.washington.edu/yenchic/20A_stat512/Lec7_Multinomial.html)
4. [Multinomial coefficients, Coastal Carolina University](https://ww2.coastal.edu/dweathers/edm/multinomials.html)
5. [MATH 465 Combinatorics lecture notes, University of Chicago](https://math.uchicago.edu/~alephnil/notes/MATH-465-notes.pdf)
6. [Neyman lecture notes on the multinomial theorem's history (der polynomische Lehrsatz)](https://veridicaldatasciencebook.stat.berkeley.edu/~census/neylogit.pdf)
7. [Abraham de Moivre, MacTutor History of Mathematics (School of Mathematics and Statistics, University of St Andrews)](https://mathshistory.st-andrews.ac.uk/SH/de_moivre_sh.pdf)
8. [Multinomial Coefficient, Wolfram MathWorld](https://mathworld.wolfram.com/MultinomialCoefficient.html)
9. [MA 320 Introductory Probability, Section 3.2, University of Kentucky](https://www.ms.uky.edu/~dmu228/ma320/lectures/sec3.2.pdf)
10. [Multinomial Series, Wolfram MathWorld](https://mathworld.wolfram.com/MultinomialSeries.html)
11. [K. K. Kataria (2016). A Probabilistic Proof of the Multinomial Theorem. arXiv.](https://arxiv.org/abs/1608.01718)
12. [Theorems and lemmas in mathematics, Chapter 1 Algebra, Theorem 22](https://amiso.my/assets/files/ysc/Theorems_and_lemmas_in_mathematics.pdf)
13. [Encyclopedia of Mathematics and Society (Greenwald and Thomley, Salem Press, 2012)](https://nzdr.ru/data/media/biblio/kolxoz/M/MGe/Greenwald%20S.J.%2C%20Thomley%20J.E.%20Encyclopedia%20of%20mathematics%20and%20society.%20Vol.1-3%20%28Salem%2C%202012%29%28O%29%28ISBN%209781587658440%29%281252s%29_MGe_.pdf)
14. [Bibliography, Series and Products in the Development of Mathematics, Cambridge University Press](https://www.cambridge.org/core/books/abs/series-and-products-in-the-development-of-mathematics/bibliography/A4F426D10F2E22AAA62ADA73C0248CB3)
15. [Proving the Binomial Theorem in Britain, 1750–1830, British Journal for the History of Mathematics](https://www.tandfonline.com/doi/full/10.1080/26375451.2026.2648415)
16. [1911 Encyclopædia Britannica, entry "Binomial", Wikisource](https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Binomial)
17. [DLMF §26.16 Multiset Permutations, NIST](https://dlmf.nist.gov/26.16)
18. [Noncommutative binomial theorem, shuffle type polynomials and Bell polynomials, arXiv](https://arxiv.org/abs/2304.06432)
19. [An elliptic extension of the multinomial theorem, arXiv](https://arxiv.org/abs/2307.12921)
20. [Efficient Methods for Determining Coefficients in Multinomial Expansions, ResearchGate](https://www.researchgate.net/publication/402709586_Efficient_Methods_for_Determining_Coefficients_in_Multinomial_Expansions)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Number theory › Elementary number theory › Combinatorial and additive number theory (elementary)*

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