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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 ...
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 ...
Beta negative binomial distribution; Bhargava factorial; Binomial (polynomial) Binomial approximation; Binomial coefficient; Binomial distribution; Binomial regression; Binomial series; Binomial theorem; Binomial transform; Binomial type; Brocard's problem
Relationship to the binomial theorem [ edit ] The Leibniz rule bears a strong resemblance to the binomial theorem , and in fact the binomial theorem can be proven directly from the Leibniz rule by taking f ( x ) = e a x {\displaystyle f(x)=e^{ax}} and g ( x ) = e b x , {\displaystyle g(x)=e^{bx},} which gives
In mathematics, the Gaussian binomial coefficients ... Like the usual binomial theorem, this formula has numerous generalizations and extensions; one such ...
The numerator is p factorial(!), which is divisible by p. However, when 0 < n < p, both n! and (p − n)! are coprime with p since all the factors are less than p and p is prime. Since a binomial coefficient is always an integer, the nth binomial coefficient is divisible by p and hence equal to 0 in the ring.
In consequence of this definition the binomial theorem can be stated by saying that the sequence {: =,,, …} is of binomial type.; The sequence of "lower factorials" is defined by = () (+). (In the theory of special functions, this same notation denotes upper factorials, but this present usage is universal among combinatorialists.)
A binomial is a polynomial which is the sum of two monomials. A binomial in a single indeterminate (also known as a univariate binomial) can be written in the form , where a and b are numbers, and m and n are distinct non-negative integers and x is a symbol which is called an indeterminate or, for historical reasons, a variable.