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.
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 |
generalized binomial theorem
ⓘ
surface form:
Newton's 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 identity
ⓘ
surface form:
Pascal's rule
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 ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.