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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 ...
A Binomial distributed random variable X ~ B(n, p) can be considered as the sum of n Bernoulli distributed random variables. So the sum of two Binomial distributed random variables X ~ B(n, p) and Y ~ B(m, p) is equivalent to the sum of n + m Bernoulli distributed random variables, which means Z = X + Y ~ B(n + m, p). This can also be proven ...
1 Historical notes. ... Download as PDF; ... Leibniz wrote to one of the Bernoulli brothers describing the similarity between the binomial theorem and the Leibniz ...
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
This list of mathematical series contains formulae for finite and infinite sums. It can be used in conjunction with other tools for evaluating sums. Here, is taken to have the value
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 .
The binomial coefficients can be arranged to form Pascal's triangle, in which each entry is the sum of the two immediately above. Visualisation of binomial expansion up to the 4th power. In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem.
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 ]