Pascal's identity
E27128
Pascal's identity is a fundamental combinatorial formula that relates adjacent binomial coefficients and underlies many proofs and properties of binomial expansions.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Pascal's rule | 3 |
| Pascal's identity canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T209692 — 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: Pascal's identity Context triple: [binomial theorem, canBeProvedBy, Pascal's identity]
-
A.
Pascal's triangle
Pascal's triangle is a triangular array of numbers in which each entry is the sum of the two directly above it, widely used in combinatorics, algebra, and probability.
-
B.
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.
-
C.
multinomial theorem
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.
-
D.
Jakob Bernoulli
Jakob Bernoulli was a pioneering Swiss mathematician of the late 17th century, renowned for his foundational work in calculus and probability theory, including the early formulation of the law of large numbers.
-
E.
Minkowski inequality
The Minkowski inequality is a fundamental result in functional analysis and measure theory that generalizes the triangle inequality to L^p spaces, providing a key tool for studying norms and integrable functions.
- 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: Pascal's identity Target entity description: Pascal's identity is a fundamental combinatorial formula that relates adjacent binomial coefficients and underlies many proofs and properties of binomial expansions.
-
A.
Pascal's triangle
Pascal's triangle is a triangular array of numbers in which each entry is the sum of the two directly above it, widely used in combinatorics, algebra, and probability.
-
B.
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.
-
C.
multinomial theorem
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.
-
D.
Jakob Bernoulli
Jakob Bernoulli was a pioneering Swiss mathematician of the late 17th century, renowned for his foundational work in calculus and probability theory, including the early formulation of the law of large numbers.
-
E.
Minkowski inequality
The Minkowski inequality is a fundamental result in functional analysis and measure theory that generalizes the triangle inequality to L^p spaces, providing a key tool for studying norms and integrable functions.
- F. None of above. chosen
Statements (42)
| Predicate | Object |
|---|---|
| instanceOf |
binomial coefficient identity
ⓘ
combinatorial identity ⓘ |
| appliesTo | combinations without order ⓘ |
| constraintOnVariables |
k ≤ n in the form C(n,k) = C(n-1,k-1) + C(n-1,k)
ⓘ
k ≥ 1 in the form C(n,k) = C(n-1,k-1) + C(n-1,k) ⓘ |
| domain |
discrete mathematics
ⓘ
probability theory ⓘ |
| equivalentTo | definition of Pascal's triangle by row recursion ⓘ |
| expresses | recursive definition of binomial coefficients ⓘ |
| hasAlternativeForm |
(n choose k) = (n-1 choose k-1) + (n-1 choose k)
ⓘ
(n+1 choose k) = (n choose k-1) + (n choose k) ⓘ C(n+1,k) = C(n,k-1) + C(n,k) ⓘ |
| hasBoundaryCondition |
C(n,0) = 1
ⓘ
C(n,n) = 1 ⓘ |
| hasFormula | C(n,k) = C(n-1,k-1) + C(n-1,k) ⓘ |
| hasGeneralization |
multinomial identities
ⓘ
q-binomial identities ⓘ |
| hasProofMethod |
algebraic manipulation of binomial coefficients
ⓘ
combinatorial argument ⓘ generating functions ⓘ |
| hasRole | fundamental identity in combinatorics ⓘ |
| hasType | linear recurrence ⓘ |
| holdsFor |
integers k with 0 ≤ k ≤ n
ⓘ
integers n ≥ 1 ⓘ |
| impliesProperty |
each entry of Pascal's triangle equals the sum of the two entries above it
ⓘ
symmetry of Pascal's triangle along its vertical axis (together with boundary conditions) ⓘ |
| involvesConcept |
Pascal's triangle
ⓘ
binomial coefficient ⓘ binomial theorem ⓘ combinatorics ⓘ |
| namedAfter | Blaise Pascal ⓘ |
| relatedTo |
Pascal's identity
self-linksurface differs
ⓘ
surface form:
Pascal's rule
Vandermonde's identity ⓘ binomial recurrence relation ⓘ |
| relatesObject | adjacent binomial coefficients ⓘ |
| usedIn |
analysis of binomial distributions
ⓘ
combinatorial counting arguments ⓘ computational derivation of binomial coefficients ⓘ construction of Pascal's triangle ⓘ dynamic programming algorithms for binomial coefficients ⓘ inductive proofs involving binomial coefficients ⓘ proof of the binomial theorem ⓘ |
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: Pascal's identity Description of subject: Pascal's identity is a fundamental combinatorial formula that relates adjacent binomial coefficients and underlies many proofs and properties of binomial expansions.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Pascal's rule
this entity surface form:
Pascal's rule
this entity surface form:
Pascal's rule