This article needs additional citations for verification .(June 2019) (Learn how and when to remove this template message) |

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 on n and b. For example (for *n* = 4),

- History
- Statement
- Examples
- Geometric explanation
- Binomial coefficients
- Formulae
- Combinatorial interpretation
- Proofs
- Combinatorial proof
- Inductive proof
- Generalizations
- Newton's generalized binomial theorem
- Further generalizations
- Multinomial theorem
- Multi-binomial theorem
- General Leibniz rule
- Applications
- Multiple-angle identities
- Series for e
- Probability
- In abstract algebra
- In popular culture
- See also
- Notes
- References
- Further reading
- External links

The coefficient a in the term of *ax*^{b}*y*^{c} is known as the binomial coefficient or (the two have the same value). These coefficients for varying n and b can be arranged to form Pascal's triangle. These numbers also arise in combinatorics, where gives the number of different combinations of b elements that can be chosen from an n-element set. Therefore is often pronounced as "n choose b".

Special cases of the binomial theorem were known since at least the 4th century BC when Greek mathematician Euclid mentioned the special case of the binomial theorem for exponent 2.^{ [1] }^{ [2] } There is evidence that the binomial theorem for cubes was known by the 6th century AD in India.^{ [1] }^{ [2] }

Binomial coefficients, as combinatorial quantities expressing the number of ways of selecting k objects out of n without replacement, were of interest to ancient Indian mathematicians. The earliest known reference to this combinatorial problem is the *Chandaḥśāstra* by the Indian lyricist Pingala (c. 200 BC), which contains a method for its solution.^{ [3] }^{:230} The commentator Halayudha from the 10th century AD explains this method using what is now known as Pascal's triangle.^{ [3] } By the 6th century AD, the Indian mathematicians probably knew how to express this as a quotient ,^{ [4] } and a clear statement of this rule can be found in the 12th century text *Lilavati* by Bhaskara.^{ [4] }

The first formulation of the binomial theorem and the table of binomial coefficients, to our knowledge, can be found in a work by Al-Karaji, quoted by Al-Samaw'al in his "al-Bahir".^{ [5] }^{ [6] }^{ [7] } Al-Karaji described the triangular pattern of the binomial coefficients^{ [8] } and also provided a mathematical proof of both the binomial theorem and Pascal's triangle, using an early form of mathematical induction.^{ [8] } The Persian poet and mathematician Omar Khayyam was probably familiar with the formula to higher orders, although many of his mathematical works are lost.^{ [2] } The binomial expansions of small degrees were known in the 13th century mathematical works of Yang Hui ^{ [9] } and also Chu Shih-Chieh.^{ [2] } Yang Hui attributes the method to a much earlier 11th century text of Jia Xian, although those writings are now also lost.^{ [3] }^{:142}

In 1544, Michael Stifel introduced the term "binomial coefficient" and showed how to use them to express in terms of , via "Pascal's triangle".^{ [10] } Blaise Pascal studied the eponymous triangle comprehensively in his *Traité dus triangle arithmétique*.^{ [11] } However, the pattern of numbers was already known to the European mathematicians of the late Renaissance, including Stifel, Niccolò Fontana Tartaglia, and Simon Stevin.^{ [10] }

Isaac Newton is generally credited with the generalized binomial theorem, valid for any rational exponent.^{ [10] }^{ [12] }

According to the theorem, it is possible to expand any nonnegative power of *x* + *y* into a sum of the form

where is an integer and each is a positive integer known as a binomial coefficient. (When an exponent is zero, the corresponding power expression is taken to be 1 and this multiplicative factor is often omitted from the term. Hence one often sees the right hand side written as .) This formula is also referred to as the **binomial formula** or the **binomial identity**. Using summation notation, it can be written as

The final expression follows from the previous one by the symmetry of x and y in the first expression, and by comparison it follows that the sequence of binomial coefficients in the formula is symmetrical. A simple variant of the binomial formula is obtained by substituting 1 for y, so that it involves only a single variable. In this form, the formula reads

or equivalently

Here are the first few cases of the binomial theorem:

In general, for the expansion of (*x* + *y*)^{n} on the right side in the nth row (numbered so that the top row is the 0th row):

- the exponents of x in the terms are
*n*,*n*−1, ..., 2, 1, 0 (the last term implicitly contains*x*^{0}= 1); - the exponents of y in the terms are 0, 1, 2, ...,
*n*−1,*n*(the first term implicitly contains*y*^{0}= 1); - the coefficients form the nth row of Pascal's triangle;
- before combining like terms, there are 2
^{n}terms*x*^{i}*y*^{j}in the expansion (not shown); - after combining like terms, there are
*n*+ 1 terms, and their coefficients sum to 2^{n}.

An example illustrating the last two points:

with .

A simple example with a specific positive value of *y*:

A simple example with a specific negative value of *y*:

For positive values of a and b, the binomial theorem with *n* = 2 is the geometrically evident fact that a square of side *a* + *b* 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 *a* + *b* 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 calculus, this picture also gives a geometric proof of the derivative ^{ [13] } if one sets and interpreting b as an infinitesimal change in a, then this picture shows the infinitesimal change in the volume of an n-dimensional hypercube, where the coefficient of the linear term (in ) is the area of the n faces, each of dimension *n*− 1:

Substituting this into the definition of the derivative via a difference quotient and taking limits means that the higher order terms, and higher, become negligible, and yields the formula interpreted as

- "the infinitesimal rate of change in volume of an n-cube as side length varies is the area of n of its (
*n*− 1)-dimensional faces".

If one integrates this picture, which corresponds to applying the fundamental theorem of calculus, one obtains Cavalieri's quadrature formula, the integral – see proof of Cavalieri's quadrature formula for details.^{ [13] }

The coefficients that appear in the binomial expansion are called **binomial coefficients**. These are usually written and pronounced "n choose k".

The coefficient of *x*^{n−k}*y*^{k} is given by the formula

which is defined in terms of the factorial function *n*!. Equivalently, this formula can be written

with k factors in both the numerator and denominator of the fraction. Although this formula involves a fraction, the binomial coefficient is actually an integer.

The binomial coefficient can be interpreted as the number of ways to choose k elements from an n-element set. This is related to binomials for the following reason: if we write (*x* + *y*)^{n} as a product

then, according to the distributive law, there will be one term in the expansion for each choice of either x or y from each of the binomials of the product. For example, there will only be one term *x*^{n}, corresponding to choosing x from each binomial. However, there will be several terms of the form *x*^{n−2}*y*^{2}, one for each way of choosing exactly two binomials to contribute a y. Therefore, after combining like terms, the coefficient of *x*^{n−2}*y*^{2} will be equal to the number of ways to choose exactly 2 elements from an n-element set.

The coefficient of *xy*^{2} in

equals because there are three *x*,*y* strings of length 3 with exactly two ys, namely,

corresponding to the three 2-element subsets of {1, 2, 3}, namely,

where each subset specifies the positions of the y in a corresponding string.

Expanding (*x* + *y*)^{n} yields the sum of the 2^{n} products of the form *e*_{1}*e*_{2} ... *e*_{n} where each *e*_{i} is *x* or y. Rearranging factors shows that each product equals *x*^{n−k}*y*^{k} for some k between 0 and n. For a given k, the following are proved equal in succession:

- the number of copies of
*x*^{n−k}*y*^{k}in the expansion - the number of n-character
*x*,*y*strings having y in exactly k positions - the number of k-element subsets of {1, 2, ...,
*n*} - either by definition, or by a short combinatorial argument if one is defining as

This proves the binomial theorem.

Induction yields another proof of the binomial theorem. When *n* = 0, both sides equal 1, since *x*^{0} = 1 and Now suppose that the equality holds for a given n; we will prove it for *n* + 1. For *j*, *k* ≥ 0, let [*f*(*x*, *y*)]_{j,k} denote the coefficient of *x*^{j}*y*^{k} in the polynomial *f*(*x*, *y*). By the inductive hypothesis, (*x* + *y*)^{n} is a polynomial in x and y such that [(*x* + *y*)^{n}]_{j,k} is if *j* + *k* = *n*, and 0 otherwise. The identity

shows that (*x* + *y*)^{n+1} is also a polynomial in x and y, and

since if *j* + *k* = *n* + 1, then (*j* − 1) + *k* = *n* and *j* + (*k* − 1) = *n*. Now, the right hand side is

by Pascal's identity.^{ [14] } On the other hand, if *j* + *k* ≠ *n* + 1, then (*j* – 1) + *k* ≠ *n* and *j* + (*k* – 1) ≠ *n*, so we get 0 + 0 = 0. Thus

which is the inductive hypothesis with *n* + 1 substituted for n and so completes the inductive step.

Around 1665, Isaac Newton generalized the binomial theorem to allow real exponents other than nonnegative integers. (The same generalization also applies to complex exponents.) In this generalization, the finite sum is replaced by an infinite series. In order to do this, one needs to give meaning to binomial coefficients with an arbitrary upper index, which cannot be done using the usual formula with factorials. However, for an arbitrary number r, one can define

where is the Pochhammer symbol, here standing for a falling factorial. This agrees with the usual definitions when r is a nonnegative integer. Then, if x and y are real numbers with |*x*| > |*y*|,^{ [Note 1] } and r is any complex number, one has

When r is a nonnegative integer, the binomial coefficients for *k* > *r* are zero, so this equation reduces to the usual binomial theorem, and there are at most *r* + 1 nonzero terms. For other values of r, the series typically has infinitely many nonzero terms.

For example, *r* = 1/2 gives the following series for the square root:

Taking *r* = −1, the generalized binomial series gives the geometric series formula, valid for |*x*| < 1:

More generally, with *r* = −*s*:

So, for instance, when *s* = 1/2,

The generalized binomial theorem can be extended to the case where x and y are complex numbers. For this version, one should again assume |*x*| > |*y*|^{ [Note 1] } and define the powers of *x* + *y* and x using a holomorphic branch of log defined on an open disk of radius |*x*| centered at x. The generalized binomial theorem is valid also for elements x and y of a Banach algebra as long as *xy* = *yx*, and x is invertible, and ||*y*/*x*|| < 1.

A version of the binomial theorem is valid for the following Pochhammer symbol-like family of polynomials: for a given real constant c, define and

for Then^{ [15] }

The case *c* = 0 recovers the usual binomial theorem.

More generally, a sequence of polynomials is said to be **binomial** if

- for all ,
- , and
- for all , , and .

An operator on the space of polynomials is said to be the *basis operator* of the sequence if and for all . A sequence is binomial if and only if its basis operator is a Delta operator.^{ [16] } Writing for the shift by operator, the Delta operators corresponding to the above "Pochhammer" families of polynomials are the backward difference for , the ordinary derivative for , and the forward difference for .

The binomial theorem can be generalized to include powers of sums with more than two terms. The general version is

where the summation is taken over all sequences of nonnegative integer indices *k*_{1} through *k*_{m} such that the sum of all *k*_{i} is n. (For each term in the expansion, the exponents must add up to n). The coefficients are known as multinomial coefficients, and can be computed by the formula

Combinatorially, the multinomial coefficient counts the number of different ways to partition an n-element set into disjoint subsets of sizes *k*_{1}, ..., *k*_{m}.

When working in more dimensions, it is often useful to deal with products of binomial expressions. By the binomial theorem this is equal to

This may be written more concisely, by multi-index notation, as

The general Leibniz rule gives the nth derivative of a product of two functions in a form similar to that of the binomial theorem:^{ [17] }

Here, the superscript (*n*) indicates the nth derivative of a function. If one sets *f*(*x*) = *e*^{ax} and *g*(*x*) = *e*^{bx}, and then cancels the common factor of *e*^{(a + b)x} from both sides of the result, the ordinary binomial theorem is recovered.^{ [18] }

For the complex numbers the binomial theorem can be combined with de Moivre's formula to yield multiple-angle formulas for the sine and cosine. According to De Moivre's formula,

Using the binomial theorem, the expression on the right can be expanded, and then the real and imaginary parts can be taken to yield formulas for cos(*nx*) and sin(*nx*). For example, since

De Moivre's formula tells us that

which are the usual double-angle identities. Similarly, since

De Moivre's formula yields

In general,

and

The number e is often defined by the formula

Applying the binomial theorem to this expression yields the usual infinite series for e. In particular:

The kth term of this sum is

As *n* → ∞, the rational expression on the right approaches 1, and therefore

This indicates that e can be written as a series:

Indeed, since each term of the binomial expansion is an increasing function of n, it follows from the monotone convergence theorem for series that the sum of this infinite series is equal to e.

The binomial theorem is closely related to the probability mass function of the negative binomial distribution. The probability of a (countable) collection of independent Bernoulli trials with probability of success all not happening is

A useful upper bound for this quantity is ^{ [19] }

The binomial theorem is valid more generally for two elements *x* and *y* in a ring, or even a semiring, provided that *xy* = *yx*. For example, it holds for two *n* × *n* matrices, provided that those matrices commute; this is useful in computing powers of a matrix.^{ [20] }

The binomial theorem can be stated by saying that the polynomial sequence {1, *x*, *x*^{2}, *x*^{3}, ...} is of binomial type.

- The binomial theorem is mentioned in the Major-General's Song in the comic opera The Pirates of Penzance.
- Professor Moriarty is described by Sherlock Holmes as having written a treatise on the binomial theorem.
- The Portuguese poet Fernando Pessoa, using the heteronym Álvaro de Campos, wrote that "Newton's Binomial is as beautiful as the Venus de Milo. The truth is that few people notice it."
^{ [21] } - In the 2014 film The Imitation Game, Alan Turing makes reference to Isaac Newton's work on the Binomial Theorem during his first meeting with Commander Denniston at Bletchley Park.

In probability theory and statistics, the **binomial distribution** with parameters *n* and *p* is the discrete probability distribution of the number of successes in a sequence of *n* independent experiments, each asking a yes–no question, and each with its own Boolean-valued outcome: *success* or *failure*. A single success/failure experiment is also called a Bernoulli trial or Bernoulli experiment, and a sequence of outcomes is called a Bernoulli process; for a single trial, i.e., *n* = 1, the binomial distribution is a Bernoulli distribution. The binomial distribution is the basis for the popular binomial test of statistical significance.

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}, and is given by the formula

In mathematics, a **combination** is a selection of items from a collection, such that the order of selection does not matter. For example, given three fruits, say an apple, an orange and a pear, there are three combinations of two that can be drawn from this set: an apple and a pear; an apple and an orange; or a pear and an orange. More formally, a *k*-**combination** of a set *S* is a subset of *k* distinct elements of *S*. If the set has *n* elements, the number of *k*-combinations is equal to the binomial coefficient

A **finite difference** is a mathematical expression of the form *f* (*x* + *b*) − *f* (*x* + *a*). If a finite difference is divided by *b* − *a*, one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the numerical solution of differential equations, especially boundary value problems.

In mathematics, **Pascal's triangle** is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in India, Persia, China, Germany, and Italy.

In mathematics, **de Moivre's formula ** states that for any real number x and integer n it holds that

In mathematics, **factorization** or **factoring** consists of writing a number or another mathematical object as a product of several *factors*, usually smaller or simpler objects of the same kind. For example, 3 × 5 is a factorization of the integer 15, and (*x* – 2)(*x* + 2) is a factorization of the polynomial *x*^{2} – 4.

In physical science and mathematics, **Legendre polynomials** are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applications. They can be defined in many ways, and the various definitions highlight different aspects as well as suggest generalizations and connections to different mathematical structures and physical and numerical applications.

In mathematics, a **recurrence relation** is an equation that recursively defines a sequence or multidimensional array of values, once one or more initial terms are given; each further term of the sequence or array is defined as a function of the preceding terms.

In mathematics, the **beta function**, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral

The **Basel problem** is a problem in mathematical analysis with relevance to number theory, first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, and read on 5 December 1735 in *The Saint Petersburg Academy of Sciences*. Since the problem had withstood the attacks of the leading mathematicians of the day, Euler's solution brought him immediate fame when he was twenty-eight. Euler generalised the problem considerably, and his ideas were taken up years later by Bernhard Riemann in his seminal 1859 paper "On the Number of Primes Less Than a Given Magnitude", in which he defined his zeta function and proved its basic properties. The problem is named after Basel, hometown of Euler as well as of the Bernoulli family who unsuccessfully attacked the problem.

**Multi-index notation** is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an ordered tuple of indices.

The **binomial series** is the Taylor series for the function given by where is an arbitrary complex number. Explicitly,

In mathematics, the **Gaussian binomial coefficients** are *q*-analogs of the binomial coefficients. 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.

In mathematics the *n*th **central binomial coefficient** is the particular binomial coefficient

In mathematics, **Pascal's rule** is a combinatorial identity about binomial coefficients. It states that for positive natural numbers *n* and *k*,

In number theory, **Lucas's theorem** expresses the remainder of division of the binomial coefficient by a prime number *p* in terms of the base *p* expansions of the integers *m* and *n*.

In mathematics, a **multisection** of a power series is a new power series composed of equally spaced terms extracted unaltered from the original series. Formally, if one is given a power series

In mathematics, **Kummer's theorem** is a formula for the exponent of the highest power of a prime number *p* that divides a given binomial coefficient. In other words, it gives the *p*-adic valuation of a binomial coefficient. The theorem is named after Ernst Kummer, who proved it in a paper,.

- 1 2 Weisstein, Eric W. "Binomial Theorem".
*Wolfram MathWorld*. - 1 2 3 4 Coolidge, J. L. (1949). "The Story of the Binomial Theorem".
*The American Mathematical Monthly*.**56**(3): 147–157. doi:10.2307/2305028. JSTOR 2305028. - 1 2 3 Jean-Claude Martzloff; S.S. Wilson; J. Gernet; J. Dhombres (1987).
*A history of Chinese mathematics*. Springer. - 1 2 Biggs, N. L. (1979). "The roots of combinatorics".
*Historia Math*.**6**(2): 109–136. doi:10.1016/0315-0860(79)90074-0. - ↑ "THE BINOMIAL THEOREM : A WIDESPREAD CONCEPT IN MEDIEVAL ISLAMIC MATHEMATICS" (PDF).
*core.ac.uk*. p. 401. Retrieved 2019-01-08. - ↑ "Taming the unknown. A history of algebra from antiquity to the early twentieth century" (PDF).
*Bulletin of the American Mathematical Society*: 727.However, algebra advanced in other respects. Around 1000, al-Karaji stated the binomial theorem

- ↑ Rashed, R. (1994-06-30).
*The Development of Arabic Mathematics: Between Arithmetic and Algebra*. Springer Science & Business Media. p. 63. ISBN 9780792325659. - 1 2 O'Connor, John J.; Robertson, Edmund F., "Abu Bekr ibn Muhammad ibn al-Husayn Al-Karaji",
*MacTutor History of Mathematics archive*, University of St Andrews . - ↑ Landau, James A. (1999-05-08). "Historia Matematica Mailing List Archive: Re: [HM] Pascal's Triangle" (mailing list email).
*Archives of Historia Matematica*. Retrieved 2007-04-13. - 1 2 3 Kline, Morris (1972).
*History of mathematical thought*. Oxford University Press. p. 273. - ↑ Katz, Victor (2009). "14.3: Elementary Probability".
*A History of Mathematics: An Introduction*. Addison-Wesley. p. 491. ISBN 0-321-38700-7. - ↑ Bourbaki, N. (18 November 1998).
*Elements of the History of Mathematics Paperback*. J. Meldrum (Translator). ISBN 978-3-540-64767-6. - 1 2 Barth, Nils R. (2004). "Computing Cavalieri's Quadrature Formula by a Symmetry of the
*n*-Cube".*The American Mathematical Monthly*.**111**(9): 811–813. doi:10.2307/4145193. ISSN 0002-9890. JSTOR 4145193, author's copy, further remarks and resources CS1 maint: postscript (link) - ↑ Binomial theorem – inductive proofs Archived February 24, 2015, at the Wayback Machine
- ↑ Sokolowsky, Dan; Rennie, Basil C. (February 1979). "Problem 352" (PDF).
*Crux Mathematicorum*.**5**(2): 55–56. - ↑ Aigner, Martin (1997) [Reprint of the 1979 Edition].
*Combinatorial Theory*. Springer. p. 105. ISBN 3-540-61787-6. - ↑ Olver, Peter J. (2000).
*Applications of Lie Groups to Differential Equations*. Springer. pp. 318–319. ISBN 9780387950006. - ↑ Spivey, Michael Z. (2019).
*The Art of Proving Binomial Identities*. CRC Press. p. 71. ISBN 978-1351215800. - ↑ Cover, Thomas M.; Thomas, Joy A. (2001-01-01).
*Data Compression*. John Wiley & Sons, Inc. p. 320. doi:10.1002/0471200611.ch5. ISBN 9780471200611. - ↑ Artin,
*Algebra*, 2nd edition, Pearson, 2018, equation (4.7.11). - ↑ "Arquivo Pessoa: Obra Édita - O binómio de Newton é tão belo como a Vénus de Milo". arquivopessoa.net.

- Bag, Amulya Kumar (1966). "Binomial theorem in ancient India".
*Indian J. History Sci*.**1**(1): 68–74. - Graham, Ronald; Knuth, Donald; Patashnik, Oren (1994). "(5) Binomial Coefficients".
*Concrete Mathematics*(2nd ed.). Addison Wesley. pp. 153–256. ISBN 978-0-201-55802-9. OCLC 17649857.

The Wikibook Combinatorics has a page on the topic of: The Binomial Theorem |

- Solomentsev, E.D. (2001) [1994], "Newton binomial",
*Encyclopedia of Mathematics*, EMS Press - Binomial Theorem by Stephen Wolfram, and "Binomial Theorem (Step-by-Step)" by Bruce Colletti and Jeff Bryant, Wolfram Demonstrations Project, 2007.

*This article incorporates material from inductive proof of binomial theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.*

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.