In mathematics, binomial coefficients are a family of positive integers that occur as coefficients in the binomial theorem.

In elementary algebra, the binomial theorem describes the algebraic expansion of powers of a binomial.

Greek mathematician Euclid mentioned the special case of the binomial theorem for exponent 2 as did the 3rd century B. C.

A more general binomial theorem and the so-called " Pascal's triangle " were known in the 10th-century A. D. to Indian mathematician Halayudha and Persian mathematician Al-Karaji, in the 11th century to Persian poet and mathematician Omar Khayyam, and in the 13th century to Chinese mathematician Yang Hui, who all derived similar results .< ref > Al-Karaji also provided a mathematical proof of both the binomial theorem and Pascal's triangle, using mathematical induction.

The most basic example of the binomial theorem is the formula for the square of x + y:

The binomial theorem can be applied to the powers of any binomial.

For a binomial involving subtraction, the theorem can be applied as long as the opposite of the second term is used.

For positive values of a and b, the binomial theorem with n = 2 is the geometrically evident fact that a square of side can be cut into a square of side a, a square of side b, and two rectangles with sides a and b. With n = 3, the theorem states that a cube of side can be cut into a cube of side a, a cube of side b, three a × a × b rectangular boxes, and three a × b × b rectangular boxes.

In Opus novum de proportionibus he introduced the binomial coefficients and the binomial theorem.

An implicit proof by mathematical induction for arithmetic sequences was introduced in the al-Fakhri written by al-Karaji around 1000 AD, who used it to prove the binomial theorem and properties of Pascal's triangle.

Abel gave a proof of the binomial theorem valid for all numbers, extending Euler's result which had held only for rationals.

( following from it, and corresponding to the binomial theorem ), are included in the observations which matured to the system of the umbral calculus.

Several theorems related to the triangle were known, including the binomial theorem.

This is the binomial theorem.

To see how the binomial theorem relates to the simple construction of Pascal's triangle, consider the problem of calculating the coefficients of the expansion of ( x + 1 )< sup > n + 1 </ sup > in terms of the corresponding coefficients of ( x + 1 )< sup > n </ sup > ( setting y

It is not difficult to turn this argument into a proof ( by mathematical induction ) of the binomial theorem.

We devote a chapter to the binomial distribution not only because it is a mathematical model for an enormous variety of real life phenomena, but also because it has important properties that recur in many other probability models.

Note that t < sup > 0 </ sup > = 1, ( 1 − t )< sup > 0 </ sup > = 1, and that the binomial coefficient,, also expressed as or is:

The same coefficient also occurs ( if ) in the binomial formula

Carl Linnaeus ( Swedish original name Carl Nilsson Linnæus, 23 May 1707 – 10 January 1778 ), also known after his ennoblement as, was a Swedish botanist, physician, and zoologist, who laid the foundations for the modern scheme of binomial nomenclature.

Flax ( also known as common flax or linseed ) ( binomial name: Linum usitatissimum ) is a member of the genus Linum in the family Linaceae.

Such a name is called a binomial name ( which may be shortened to just " binomial "), a binomen or a scientific name ; more informally it is also called a Latin name.

The binomial coefficient is then defined by the multiplicative formula and can also be rewritten using the gamma function:

: The blue picture illustrates an example of fitting the exponential distribution to ranked annually maximum one-day rainfalls showing also the 90 % confidence belt based on the binomial distribution.

If x, and therefore also cos x and sin x, are real numbers, then the identity of these parts can be written using binomial coefficients.

Bernoulli trials may also lead to negative binomial distributions ( which count the number of successes in a series of repeated Bernoulli trials until a specified number of failures are seen ), as well as various other distributions.

Mate plant, or Yerba mate (; also spelled in English as maté, from the,, ), binomial name Ilex paraguariensis, is a species of holly ( family Aquifoliaceae ), well known as the source of the mate beverage.

For the special case of, there is a closely related set of polynomials, also called the Newton polynomials, that are simply the binomial coefficients for general argument.

The bean machine, also known as the quincunx or Galton box, is a device invented by Sir Francis Galton to demonstrate the central limit theorem, in particular that the normal distribution is approximate to the binomial distribution.

Monte Carlo simulations are also less susceptible to sampling errors, since binomial techniques use discrete time units.

In computer science, a binomial heap is a heap similar to a binary heap but also supports quick merging of two heaps.

The blue picture illustrates an example of fitting the Gumbel distribution to ranked maximum one-day October rainfalls showing also the 90 % confidence belt based on the binomial distribution.

He also created the binomial theorem, worked extensively on optics, and created a law of cooling.

The Touchard polynomials, studied by, also called the exponential polynomials, comprise a polynomial sequence of binomial type defined by

This expression can also be given in the form of a binomial coefficient, as a polynomial expression in d, or using a rising factorial power of:

Since is the digit repeated n times ( because, see also binomial numbers ), the diminished radix complement of a number is found by complementing each digit with respect to ( that is, subtracting each digit in y from ).

The Mellin transform also connects the Newton series or binomial transform together with the Poisson generating function, by means of the Poisson – Mellin – Newton cycle.

which holds for general complex-valued s, for the suitably extended binomial coefficients.

A similar argument holds for merging the trees in the data structures discussed below, additionally it helps explain the time analysis of some operations in the binomial heap and Fibonacci heap data structures.

holds, where the parentheses denote a binomial coefficient.

We now generalize these ideas for general binomial experiments.

`` Success '' and `` failure '' are just convenient labels for the two categories of outcomes when we talk about binomial trials in general.

Note that we need not know the value of p, for the experiment to be binomial.

Strictly speaking, this means that the probability for each possible outcome of the experiment can be computed by multiplying together the probabilities of the possible outcomes of the single binomial trials.

When this experiment is viewed as composed of five binomial trials, one for each member of the family, the outcomes of the trials are obviously not independent.

Indeed, the experiment is better viewed as consisting of one binomial trial for the entire family.

) On the other hand, even when the binomial model does not describe well the physical phenomenon being studied, the binomial model may still be used as a baseline for comparative purposes ; ;

Binomial distributions were treated by James Bernoulli about 1700, and for this reason binomial trials are sometimes called Bernoulli trials.

His Traité du triangle arithmétique (" Treatise on the Arithmetical Triangle ") of 1653 described a convenient tabular presentation for binomial coefficients, now called Pascal's triangle.

In 1754 Carl von Linné ( Carl Linnaeus ) divided the plant Kingdom into 25 classes in a taxonomy with a standardized binomial naming system for animal and plant species.

Arranging binomial coefficients into rows for successive values of n, and in which k ranges from 0 to n, gives a triangular array called Pascal's triangle.

This shows in particular that is a natural number for any natural numbers n and k. 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 < sub > i </ sub > is a nonnegative integer is given by.

One has a recursive formula for binomial coefficients

A variant of the binomial formula is obtained by substituting 1 for y, so that it involves only a single variable.