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In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial.According to the theorem, the power (+) expands into a polynomial with terms of the form , where the exponents and are nonnegative integers satisfying + = and the coefficient of each term is a specific positive integer ...
The solution to this particular problem is given by the binomial coefficient (+), which is the number of subsets of size k − 1 that can be formed from a set of size n + k − 1. If, for example, there are two balls and three bins, then the number of ways of placing the balls is ( 2 + 3 − 1 3 − 1 ) = ( 4 2 ) = 6 {\displaystyle {\tbinom {2 ...
If is an odd prime and <, then (,) =. Proof: There are exactly two factors of in the numerator of the expression () = ()! / (!), coming from the two terms and in ()!, and also two factors of in the denominator from one copy of the term in each of the two factors of !.
ATS theorem (number theory) Abel's binomial theorem (combinatorics) Abel's curve theorem (mathematical analysis) Abel's theorem (mathematical analysis) Abelian and Tauberian theorems (mathematical analysis) Abel–Jacobi theorem (algebraic geometry) Abel–Ruffini theorem (theory of equations, Galois theory) Abhyankar–Moh theorem (algebraic ...
Binomial (polynomial), a polynomial with two terms; Binomial coefficient, numbers appearing in the expansions of powers of binomials; Binomial QMF, a perfect-reconstruction orthogonal wavelet decomposition; Binomial theorem, a theorem about powers of binomials; Binomial type, a property of sequences of polynomials; Binomial series, a ...
In mathematics, the multinomial theorem describes how to expand a power of a sum in terms of powers of the terms in that sum. It is the generalization of the binomial theorem from binomials to multinomials .
The binomial approximation for the square root, + + /, can be applied for the following expression, + where and are real but .. The mathematical form for the binomial approximation can be recovered by factoring out the large term and recalling that a square root is the same as a power of one half.
The binomial coefficients can be arranged to form Pascal's triangle, in which each entry is the sum of the two immediately above. Visualisation of binomial expansion up to the 4th power. In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem.