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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 central binomial coefficients give the number of possible number of assignments of n-a-side sports teams from 2n players, taking into account the playing area side. The central binomial coefficient () is the number of arrangements where there are an equal number of two types of objects.
Like the usual binomial theorem, this formula has numerous generalizations and extensions; one such, corresponding to Newton's generalized binomial theorem for negative powers, is ∏ k = 0 n − 1 1 1 − q k t = ∑ k = 0 ∞ ( n + k − 1 k ) q t k . {\displaystyle \prod _{k=0}^{n-1}{\frac {1}{1-q^{k}t}}=\sum _{k=0}^{\infty }{n+k-1 \choose k ...
This proof of the multinomial theorem uses the binomial theorem and induction on m.. First, for m = 1, both sides equal x 1 n since there is only one term k 1 = n in the sum. For the induction step, suppose the multinomial theorem holds for m.
The word FOIL is an acronym for the four terms of the product: First ("first" terms of each binomial are multiplied together) Outer ("outside" terms are multiplied—that is, the first term of the first binomial and the second term of the second) Inner ("inside" terms are multiplied—second term of the first binomial and first term of the second)
In mathematics, Pascal's rule (or Pascal's formula) is a combinatorial identity about binomial coefficients.It states that for positive natural numbers n and k, + = (), where () is a binomial coefficient; one interpretation of the coefficient of the x k term in the expansion of (1 + x) n.
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