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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, it is possible to expand the polynomial (x + y) n into a sum involving terms of the form ax b y c, where the exponents b and c are nonnegative integers with b + c = n, and the coefficient a of each term is a specific positive integer depending ...
Commonly, a binomial coefficient is indexed by a pair of integers n ≥ k ≥ 0 and is written It is the coefficient of the xk term in the polynomial expansion of the binomial power (1 + x)n; this coefficient can be computed by the multiplicative formula. which using factorial notation can be compactly expressed as.
Proof of Bertrand's postulate. In mathematics, Bertrand's postulate (now a theorem) states that, for each , there is a prime such that . First conjectured in 1845 by Joseph Bertrand, [1] it was first proven by Chebyshev, and a shorter but also advanced proof was given by Ramanujan. [2]
In a simple symmetric random walk on a locally finite lattice, the probabilities of the location jumping to each one of its immediate neighbors are the same. The best-studied example is the random walk on the d -dimensional integer lattice (sometimes called the hypercubic lattice) Z d {\displaystyle \mathbb {Z} ^{d}} .
Vandermonde's identity. In combinatorics, Vandermonde's identity (or Vandermonde's convolution) is the following identity for binomial coefficients: for any nonnegative integers r, m, n. The identity is named after Alexandre-Théophile Vandermonde (1772), although it was already known in 1303 by the Chinese mathematician Zhu Shijie. [1]
Buffon's needle was the earliest problem in geometric probability to be solved; [2] it can be solved using integral geometry. The solution for the sought probability p, in the case where the needle length l is not greater than the width t of the strips, is. This can be used to design a Monte Carlo method for approximating the number π ...
The binomial distribution is frequently used to model the number of successes in a sample of size n drawn with replacement from a population of size N. If the sampling is carried out without replacement, the draws are not independent and so the resulting distribution is a hypergeometric distribution, not a binomial one.
The binomial approximation is useful for approximately calculating powers of sums of 1 and a small number x. It states that. It is valid when and where and may be real or complex numbers. The benefit of this approximation is that is converted from an exponent to a multiplicative factor. This can greatly simplify mathematical expressions (as in ...