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.
Aliases (1)
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 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 → |
Referenced by (3)
| Subject (surface form when different) | Predicate |
|---|---|
|
binomial theorem
→
|
canBeProvedBy |
|
binomial theorem
("Pascal's rule")
→
|
relatedConcept |
|
Pascal's identity
("Pascal's rule")
→
|
relatedTo |