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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 method relies on two observations. First, many identities can be proved by extracting coefficients of generating functions. Second, many generating functions are convergent power series, and coefficient extraction can be done using the Cauchy residue theorem (usually this is done by integrating over a small circular contour enclosing the ...
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 ...
Multinomial coefficient as a product of binomial coefficients, counting the permutations of the letters of MISSISSIPPI. The multinomial coefficient (, …,) is also the number of distinct ways to permute a multiset of n elements, where k i is the multiplicity of each of the i th element. For example, the number of distinct permutations of the ...
Here, (+) is the binomial coefficient "p + 1 choose r", and the B j are the Bernoulli numbers with the convention that = +. The result: Faulhaber's formula [ edit ]
This coefficient can be found using binomial series and agrees with the result of Theorem two, namely (+). This Cauchy product expression is justified via stars and bars: the coefficient of x n {\displaystyle x^{n}} in the expansion of the product
The Gaussian binomial coefficient, written as () or [], is a polynomial in q with integer coefficients, whose value when q is set to a prime power counts the number of subspaces of dimension k in a vector space of dimension n over , a finite field with q elements; i.e. it is the number of points in the finite Grassmannian (,).
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).