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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| multinomial theorem canonical | 2 |
| Pascal's pyramid | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T209665 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: multinomial theorem Context triple: [binomial theorem, isSpecialCaseOf, multinomial theorem]
-
A.
binomial theorem
The binomial theorem is a fundamental algebraic formula that provides a systematic way to expand powers of binomial expressions, playing a key role in combinatorics and mathematical analysis.
-
B.
central limit theorem
The central limit theorem is a fundamental result in probability theory stating that the sum (or average) of many independent, identically distributed random variables tends to follow a normal distribution, regardless of the original variables’ distribution, under mild conditions.
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C.
Minkowski sum
The Minkowski sum is a fundamental operation in geometry and convex analysis that combines two sets by adding every vector in one set to every vector in the other, widely used in areas such as optimization, robotics, and computational geometry.
-
D.
Born–Huang expansion
The Born–Huang expansion is a quantum mechanical method that systematically improves upon the Born–Oppenheimer approximation by including couplings between electronic and nuclear motions in molecular systems.
-
E.
Bose–Einstein statistics
Bose–Einstein statistics is a quantum statistical framework that describes the distribution and collective behavior of indistinguishable bosons, underpinning phenomena such as Bose–Einstein condensation.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: multinomial theorem Target entity description: 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.
-
A.
binomial theorem
The binomial theorem is a fundamental algebraic formula that provides a systematic way to expand powers of binomial expressions, playing a key role in combinatorics and mathematical analysis.
-
B.
central limit theorem
The central limit theorem is a fundamental result in probability theory stating that the sum (or average) of many independent, identically distributed random variables tends to follow a normal distribution, regardless of the original variables’ distribution, under mild conditions.
-
C.
Minkowski sum
The Minkowski sum is a fundamental operation in geometry and convex analysis that combines two sets by adding every vector in one set to every vector in the other, widely used in areas such as optimization, robotics, and computational geometry.
-
D.
Born–Huang expansion
The Born–Huang expansion is a quantum mechanical method that systematically improves upon the Born–Oppenheimer approximation by including couplings between electronic and nuclear motions in molecular systems.
-
E.
Bose–Einstein statistics
Bose–Einstein statistics is a quantum statistical framework that describes the distribution and collective behavior of indistinguishable bosons, underpinning phenomena such as Bose–Einstein condensation.
- F. None of above. chosen
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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: multinomial theorem Description of subject: 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.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Pascal's pyramid