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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 ...
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 (). It is the coefficient of the x k term in the polynomial expansion of the binomial power (1 + x) n; this coefficient can be ...
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 (,).
The approximation can be proven several ways, and is closely related to the binomial theorem. By Bernoulli's inequality , the left-hand side of the approximation is greater than or equal to the right-hand side whenever x > − 1 {\displaystyle x>-1} and α ≥ 1 {\displaystyle \alpha \geq 1} .
where the power series on the right-hand side of is expressed in terms of the (generalized) binomial coefficients ():= () (+)!.Note that if α is a nonnegative integer n then the x n + 1 term and all later terms in the series are 0, since each contains a factor of (n − n).
In mathematics, the multinomial theorem describes how to expand a power of a sum in terms of powers of the terms in that sum. It is the generalization of the binomial theorem from binomials to multinomials .
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 x {\displaystyle x} can be taken to be, for example, a complex number , including negative integers, or a polynomial with complex coefficients, or any complex-valued ...
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