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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 ...
This is a list of factorial and binomial topics in mathematics. See also binomial (disambiguation). Abel's binomial theorem; Alternating factorial; Antichain; Beta function; Bhargava factorial; Binomial coefficient. Pascal's triangle; Binomial distribution; Binomial proportion confidence interval; Binomial-QMF (Daubechies wavelet filters ...
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.
A general theory covering such relations, including the falling and rising factorial functions, is given by the theory of polynomial sequences of binomial type and Sheffer sequences. Falling and rising factorials are Sheffer sequences of binomial type, as shown by the relations:
In algebra, the factorials arise through the binomial theorem, which uses binomial coefficients to expand powers of sums. [30] They also occur in the coefficients used to relate certain families of polynomials to each other, for instance in Newton's identities for symmetric polynomials. [31]
The binomial coefficients are all integers. The numerator contains a factor p by the definition of factorial. When 0 < i < p, neither of the terms in the denominator includes a factor of p (relying on the primality of p), leaving the coefficient itself to possess a prime factor of p from the numerator, implying that
Pages in category "Factorial and binomial topics" ... Binomial theorem; Binomial transform; Binomial type; Brocard's problem; C. Carlson's theorem; Catalan number;
A binomial raised to the n th power, represented as (x + y) n can be expanded by means of the binomial theorem or, equivalently, using Pascal's triangle. For example, the square (x + y) 2 of the binomial (x + y) is equal to the sum of the squares of the two terms and twice the product of the terms, that is: