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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}}.}
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 ...
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 mathematics, Pascal's triangle is an infinite triangular array of the binomial coefficients which play a crucial role in probability theory, combinatorics, and algebra.In much of the Western world, it is named after the French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in Persia, [1] India, [2] China, Germany, and Italy.
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).
Here, (+) is the binomial coefficient "p + 1 choose r", and the B j are the Bernoulli ... Specifically, using the binomial theorem, we get + ...
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
Using the induction hypothesis, we have that k p ≡ k (mod p); and, trivially, 1 p = 1. Thus (+) + (), which is the statement of the theorem for a = k+1. ∎ In order to prove the lemma, we must introduce the binomial theorem, which states that for any positive integer n,