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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 ...
There are many other combinatorial interpretations of binomial coefficients (counting problems for which the answer is given by a binomial coefficient expression), for instance the number of words formed of n bits (digits 0 or 1) whose sum is k is given by (), while the number of ways to write = + + + where every a i is a nonnegative integer is ...
Thus many identities on binomial coefficients carry over to the falling and rising factorials. The rising and falling factorials are well defined in any unital ring, and therefore can be taken to be, for example, a complex number, including negative integers, or a polynomial with complex coefficients, or any complex-valued function.
Abel's binomial theorem, named after Niels Henrik Abel, is a mathematical identity involving sums of binomial coefficients. It states the following: ... Example. 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).
A simple example is the Fermat factorization method, which considers the sequence of numbers :=, for := ⌈ ⌉ +. If one of the x i {\displaystyle x_{i}} equals a perfect square b 2 {\displaystyle b^{2}} , then N = a i 2 − b 2 = ( a i + b ) ( a i − b ) {\displaystyle N=a_{i}^{2}-b^{2}=(a_{i}+b)(a_{i}-b)} is a (potentially non-trivial ...
Since a binomial coefficient is always an integer, the nth binomial coefficient is divisible by p and hence equal to 0 in the ring. We are left with the zeroth and pth coefficients, which both equal 1, yielding the desired equation. Thus in characteristic p the freshman's dream is a valid identity.