multinomial theorem
E26831
The multinomial theorem is a fundamental algebraic formula that generalizes the binomial theorem to express powers of sums with any number of terms using multinomial coefficients.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| Pascal's pyramid | 1 |
Statements (44)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic identity
ⓘ
mathematical theorem ⓘ |
| appearsIn |
expansion of (a+b+c+...)^n
ⓘ
higher-dimensional Taylor expansions ⓘ moment calculations in statistics ⓘ |
| appliesTo | (x1 + x2 + ... + xm)^n ⓘ |
| assumes |
m is a positive integer
ⓘ
n is a nonnegative integer ⓘ x1,...,xm are elements of a commutative ring or field ⓘ |
| basisFor |
definition of multinomial distribution probabilities
ⓘ
some algorithms in symbolic computation ⓘ |
| category | finite sums and products identities ⓘ |
| coefficientOf | x1^{k1}...xm^{km} equals n!/(k1!...km!) ⓘ |
| connectedTo |
combinatorial counting of distributions of n identical objects into m boxes
ⓘ
stars and bars method ⓘ |
| describes | expansion of powers of sums with multiple terms ⓘ |
| field |
algebra
ⓘ
combinatorics ⓘ |
| generalizationOf |
generalized binomial theorem
ⓘ
surface form:
Newton binomial formula
|
| generalizes | binomial theorem ⓘ |
| hasComponent | multinomial coefficient n!/(k1!...km!) ⓘ |
| implies | number of distinct terms in expansion equals C(n+m-1, m-1) ⓘ |
| involves | sum over all m-tuples of nonnegative integers (k1,...,km) with k1+...+km=n ⓘ |
| proofMethods |
combinatorial argument
ⓘ
generating functions ⓘ induction on n ⓘ |
| relatedTo |
multinomial coefficient
ⓘ
multinomial distribution ⓘ |
| relates | factorials to exponents in polynomial expansions ⓘ |
| requires | commutativity of multiplication for variables x1,...,xm ⓘ |
| specialCase | reduces to binomial theorem when m = 2 ⓘ |
| states | (x1 + x2 + ... + xm)^n = Σ_{k1+...+km=n} (n!/(k1!...km!)) x1^{k1}...xm^{km} ⓘ |
| typeOfGeneralization | combinatorial generalization of binomial expansion ⓘ |
| usedFor |
computing coefficients in multivariate polynomials
ⓘ
counting ordered outcomes in m-way experiments with n trials ⓘ |
| usedIn |
combinatorial proofs
ⓘ
enumerative combinatorics ⓘ polynomial algebra ⓘ probability theory ⓘ series expansions ⓘ |
| uses | multinomial coefficients ⓘ |
| validOver |
any commutative ring with identity
ⓘ
complex numbers ⓘ real numbers ⓘ |
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Pascal's pyramid