binomial theorem
E4686
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| binomial theorem canonical | 3 |
How this entity was disambiguated
This entity first appeared as the object of triple T80199 — 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: binomial theorem Context triple: [Isaac Newton, knownFor, binomial theorem]
-
A.
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.
-
B.
Feynman–Kac formula
The Feynman–Kac formula is a fundamental result connecting solutions of certain partial differential equations with expectations over stochastic processes, forming a bridge between quantum mechanics, probability theory, and mathematical finance.
-
C.
Feynman–Hellmann theorem
The Feynman–Hellmann theorem is a result in quantum mechanics that relates the derivative of an energy eigenvalue with respect to a parameter in the Hamiltonian to the expectation value of the corresponding derivative of the Hamiltonian.
-
D.
implicit function theorem
The implicit function theorem is a fundamental result in calculus and differential geometry that guarantees, under suitable smoothness and nondegeneracy conditions, the local solvability of equations for some variables as differentiable functions of others.
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E.
Einstein coefficients
Einstein coefficients are parameters in quantum theory that quantify the probabilities of absorption, spontaneous emission, and stimulated emission of radiation by atoms or molecules.
- 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: binomial theorem Target entity description: 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.
-
A.
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.
-
B.
Feynman–Kac formula
The Feynman–Kac formula is a fundamental result connecting solutions of certain partial differential equations with expectations over stochastic processes, forming a bridge between quantum mechanics, probability theory, and mathematical finance.
-
C.
Feynman–Hellmann theorem
The Feynman–Hellmann theorem is a result in quantum mechanics that relates the derivative of an energy eigenvalue with respect to a parameter in the Hamiltonian to the expectation value of the corresponding derivative of the Hamiltonian.
-
D.
implicit function theorem
The implicit function theorem is a fundamental result in calculus and differential geometry that guarantees, under suitable smoothness and nondegeneracy conditions, the local solvability of equations for some variables as differentiable functions of others.
-
E.
Einstein coefficients
Einstein coefficients are parameters in quantum theory that quantify the probabilities of absorption, spontaneous emission, and stimulated emission of radiation by atoms or molecules.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf | mathematical theorem ⓘ |
| appliesTo | (a + b)^n ⓘ |
| canBeProvedBy |
Pascal's identity
ⓘ
combinatorial arguments ⓘ mathematical induction ⓘ |
| category |
theorems in algebra
ⓘ
theorems in combinatorics ⓘ |
| defines | \binom{α}{k} = α(α-1)…(α-k+1)/k! for generalized exponents ⓘ |
| describes | expansion of powers of a binomial ⓘ |
| field |
algebra
ⓘ
combinatorics ⓘ mathematical analysis ⓘ |
| forComplexExponent | (1 + x)^α = Σ_{k=0}^∞ \binom{α}{k} x^k for |x| < 1 and α ∈ ℂ ⓘ |
| forRealExponent | (1 + x)^α = Σ_{k=0}^∞ \binom{α}{k} x^k for |x| < 1 ⓘ |
| generalizedBy | multinomial theorem ⓘ |
| hasGeneralization |
Newton's generalized binomial theorem
ⓘ
linked to:
generalized binomial theorem
generalized binomial theorem ⓘ |
| hasGeneralTerm | C(n,k) a^{n-k} b^k ⓘ |
| hasHistoricalAttribution | Isaac Newton ⓘ |
| hasNumberOfTerms | n + 1 ⓘ |
| hasSpecialCase |
(a + b)^2 = a^2 + 2ab + b^2
ⓘ
(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 ⓘ (a + b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4 ⓘ |
| implies | entries of Pascal's triangle are binomial coefficients ⓘ |
| impliesIdentity |
Σ_{k=0}^n (-1)^k C(n,k) = 0 for n > 0
ⓘ
Σ_{k=0}^n C(n,k) = 2^n ⓘ Σ_{k=0}^n k C(n,k) = n 2^{n-1} ⓘ Σ_{k=0}^n k^2 C(n,k) = n(n+1)2^{n-2} ⓘ |
| isSpecialCaseOf | multinomial theorem ⓘ |
| knownSince | at least the 17th century in its general form ⓘ |
| relatedConcept |
Pascal's rule
ⓘ
linked to:
Pascal's identity
binomial coefficient identity ⓘ |
| relatesTo | Pascal's triangle ⓘ |
| requires |
commutativity of addition for a and b in its usual form
ⓘ
n to be a nonnegative integer in its classical form ⓘ |
| states | (a + b)^n = Σ_{k=0}^n C(n,k) a^{n-k} b^k ⓘ |
| symbolicallyUses |
C(n,k)
ⓘ
\binom{n}{k} ⓘ |
| usedIn |
Taylor series computations
ⓘ
algebraic manipulation of polynomials ⓘ analysis of algorithms ⓘ binomial distribution ⓘ combinatorial counting problems ⓘ finite difference methods ⓘ probability theory ⓘ series expansions ⓘ |
| uses | binomial coefficients ⓘ |
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: binomial theorem Description of subject: 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.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.