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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 ...
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 . Commonly, a binomial coefficient is indexed by a pair of integers n ≥ k ≥ 0 and is written ( n k ) . {\displaystyle {\tbinom {n}{k}}.}
The case α = 1 gives the series 1 + x + x 2 + x 3 + ..., where the coefficient of each term of the series is simply 1. The case α = 2 gives the series 1 + 2x + 3x 2 + 4x 3 + ..., which has the counting numbers as coefficients. The case α = 3 gives the series 1 + 3x + 6x 2 + 10x 3 + ..., which has the triangle numbers as coefficients.
The sum of the entries is p 2 + 2pq + q 2 = 1, as the genotype frequencies must sum to one. Note again that as p + q = 1, the binomial expansion of (p + q) 2 = p 2 + 2pq + q 2 = 1 gives the same relationships.
In a 1940 article on modular fields, Saunders Mac Lane quotes Stephen Kleene's remark that a knowledge of (a + b) 2 = a 2 + b 2 in a field of characteristic 2 would corrupt freshman students of algebra. This may be the first connection between "freshman" and binomial expansion in fields of positive characteristic. [6]
In mathematics, Kummer's theorem is a formula for the exponent of the highest power of a prime number p that divides a given binomial coefficient. In other words, it gives the p-adic valuation of a binomial coefficient. The theorem is named after Ernst Kummer, who proved it in a paper, (Kummer 1852).
This section needs expansion. You can help by adding to it. (February 2024) Test name ... ≥2: Binomial test: binary: non-parametric: univariate: 1: Yes: Multinomial ...
The binomial approximation is useful for approximately calculating powers of sums of 1 and a small number x. It states that It states that ( 1 + x ) α ≈ 1 + α x . {\displaystyle (1+x)^{\alpha }\approx 1+\alpha x.}