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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 ...
Berge's theorem (graph theory) Binomial theorem (algebra, combinatorics) Bondy's theorem (graph theory, combinatorics) Bondy–Chvátal theorem (graph theory) Brooks's theorem (graph theory) Bruck–Chowla–Ryser theorem (combinatorics) Cameron–Erdős theorem (discrete mathematics) Corners theorem (arithmetic combinatorics) Courcelle's ...
The falling factorial occurs in a formula which represents polynomials using the forward difference operator = (+) , which in form is an exact analogue to Taylor's theorem: Compare the series expansion from umbral calculus
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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.
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
described in terms of the factorial function, n! = 1×2×3×⋯×n. Proof of Lemma. We consider the binomial coefficient when the exponent is a prime p: =!! ()! The binomial coefficients are all integers. The numerator contains a factor p by the definition of factorial.