Binomial theorem

In elementary algebra, the binomial theorem 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 depending on and. For example, for,
The coefficient in each term is known as the binomial coefficient or . These coefficients for varying and can be arranged to form Pascal's triangle. These numbers also occur in combinatorics, where gives the number of different combinations of elements that can be chosen from an -element set. Therefore is usually pronounced as " choose ".

Statement

According to the theorem, the expansion of any nonnegative integer power of the binomial is a sum of the form
where each is a positive integer known as a binomial coefficient, defined as
This formula is also referred to as the binomial formula or the binomial identity. Using summation notation, it can be written more concisely as
The final expression follows from the previous one by the symmetry of and in the first expression, and by comparison it follows that the sequence of binomial coefficients in the formula is symmetric,
A simple variant of the binomial formula is obtained by substituting for, so that it involves only a single variable. In this form, the formula reads

Examples

The first few cases of the binomial theorem are:
In general, for the expansion of on the right side in the th row :

the exponents of in the terms are ;
the exponents of in the terms are ;
the coefficients form the th row of Pascal's triangle;
before combining like terms, there are terms in the expansion ;
after combining like terms, there are terms, and their coefficients sum to.

An example illustrating the last two points: with.
A simple example with a specific positive value of :
A simple example with a specific negative value of :

Geometric explanation

For positive values of and, the binomial theorem with is the geometrically evident fact that a square of side can be cut into a square of side, a square of side, and two rectangles with sides and. With, the theorem states that a cube of side can be cut into a cube of side, a cube of side, three rectangular boxes, and three rectangular boxes.
In calculus, this picture also gives a geometric proof of the derivative if one sets and interpreting as an infinitesimal change in, then this picture shows the infinitesimal change in the volume of an -dimensional hypercube, where the coefficient of the linear term is the area of the faces, each of dimension :
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 -cube as side length varies is the area of of its -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.

Binomial coefficients

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

Formulas

The coefficient of is given by the formula
which is defined in terms of the factorial function. Equivalently, this formula can be written
with factors in both the numerator and denominator of the fraction. Although this formula involves a fraction, the binomial coefficient is actually an integer.

Combinatorial interpretation

The binomial coefficient can be interpreted as the number of ways to choose elements from an -element set. This is related to binomials for the following reason: if we write as a product
then, according to the distributive law, there will be one term in the expansion for each choice of either or from each of the binomials of the product. For example, there will only be one term, corresponding to choosing from each binomial. However, there will be several terms of the form, one for each way of choosing exactly two binomials to contribute a. Therefore, after combining like terms, the coefficient of will be equal to the number of ways to choose exactly elements from an -element set.

Proofs

Combinatorial proof

Expanding yields the sum of the products of the form where each is or . Rearranging factors shows that each product equals for some between and . For a given, the following are proved equal in succession:

the number of terms equal to in the expansion
the number of -character strings having in exactly positions
the number of -element subsets of
either by definition, or by a short combinatorial argument if one is defining as

This proves the binomial theorem.

Example

The coefficient of in
equals because there are three strings of length 3 with exactly two 's, namely,
corresponding to the three 2-element subsets of, namely,
where each subset specifies the positions of the in a corresponding string.

Inductive proof

yields another proof of the binomial theorem. When, both sides equal, since and Now suppose that the equality holds for a given ; we will prove it for. For, let denote the coefficient of in the polynomial. By the inductive hypothesis, is a polynomial in and such that is if, and otherwise. The identity
shows that is also a polynomial in and, and
since if, then and. Now, the right hand side is
by Pascal's identity. On the other hand, if, then and, so we get. Thus
which is the inductive hypothesis with substituted for and so completes the inductive step.

Generalizations

Generalized binomial theorem

The standard binomial theorem, as discussed above, is concerned with where the exponent is a nonnegative integer. The generalized binomial theorem allows for non-integer, negative, or even complex exponents, at the expense of replacing the finite sum 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, one can define
where the last equation introduces modern notation for the falling factorial. This agrees with the usual definitions when is a nonnegative integer. Then, if and are real numbers with, and is any complex number, one has
When is a nonnegative integer, the binomial coefficients for are zero, so this equation reduces to the usual binomial theorem, and there are at most nonzero terms. For other values of, the series has infinitely many nonzero terms.
For example, gives the following series for the square root:
With, the generalized binomial series becomes:
which is the geometric series sum formula for the convergent case, whose common ratio is.
More generally, with, we have for :
So, for instance, when,
Replacing with yields:
So, for instance, when, we have for :

Further generalizations

The generalized binomial theorem can be extended to the case where and are complex numbers. For this version, one should again assume and define the powers of and using a holomorphic branch of log defined on an open disk of radius centered at. The generalized binomial theorem is valid also for elements and of a Banach algebra as long as, and is invertible, and.
A version of the binomial theorem is valid for the following Pochhammer symbol-like family of polynomials: for a given real constant, define and
for Then
The case recovers the usual binomial theorem.
More generally, a sequence of polynomials is said to be of binomial type 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. 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.

Multinomial theorem

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 through such that the sum of all is .. 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 -element set into disjoint subsets of sizes.

Multi-binomial theorem

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

General Leibniz rule

The general Leibniz rule gives the th derivative of a product of two functions in a form similar to that of the binomial theorem:
Here, the superscript indicates the th derivative of a function,. If one sets and, cancelling the common factor of from each term gives the ordinary binomial theorem.

History

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. Greek mathematician Diophantus cubed various binomials, including. Indian mathematician Aryabhata's method for finding cube roots, from around 510 AD, suggests that he knew the binomial formula for exponent.
Binomial coefficients, as combinatorial quantities expressing the number of ways of selecting objects out of without replacement, were of interest to ancient Indian mathematicians. The Jain Bhagavati Sutra describes the number of combinations of philosophical categories, senses, or other things, with correct results up through and a suggestion that higher combinations could likewise be found. The Chandaḥśāstra by the Indian lyricist Piṅgala somewhat cryptically describes a method of arranging two types of syllables to form metres of various lengths and counting them; as interpreted and elaborated by Piṅgala's 10th-century commentator Halāyudha his "method of pyramidal expansion" for counting metres is equivalent to Pascal's triangle. Varāhamihira describes another method for computing combination counts by adding numbers in columns. By the 9th century at latest Indian mathematicians learned to express this as a product of fractions, and clear statements of this rule can be found in Śrīdhara's Pāṭīgaṇita, Mahāvīra's Gaṇita-sāra-saṅgraha, and Bhāskara II's Līlāvatī.
The Persian mathematician al-Karajī wrote a now-lost book containing the binomial theorem and a table of binomial coefficients, often credited as their first appearance.
An explicit statement of the binomial theorem appears in al-Samawʾal's al-Bāhir, there credited to al-Karajī. Al-Samawʾal algebraically expanded the square, cube, and fourth power of a binomial, each in terms of the previous power, and noted that similar proofs could be provided for higher powers, an early form of mathematical induction. He then provided al-Karajī's table of binomial coefficients up to and a rule for generating them equivalent to the recurrence relation. The Persian poet and mathematician Omar Khayyam was probably familiar with the formula to higher orders, although many of his mathematical works are lost. The binomial expansions of small degrees were known in the 13th century mathematical works of Yang Hui and also Chu Shih-Chieh. Yang Hui attributes the method to a much earlier 11th century text of Jia Xian, although those writings are now also lost.
In Europe, descriptions of the construction of Pascal's triangle can be found as early as Jordanus de Nemore's De arithmetica. In 1544, Michael Stifel introduced the term "binomial coefficient" and showed how to use them to express in terms of, via "Pascal's triangle". Other 16th century mathematicians including Niccolò Fontana Tartaglia and Simon Stevin also knew of it. 17th-century mathematician Blaise Pascal studied the eponymous triangle comprehensively in his Traité du triangle arithmétique.
The development of the binomial theorem for positive integer exponents is attributed to Al-Kashi by the year 1427. The first proper proof of the binomial theorem for positive integral index was given by Pascal. By the early 17th century, some specific cases of the generalized binomial theorem, such as for, can be found in the work of Henry Briggs' Arithmetica Logarithmica. Isaac Newton discovered the generalized binomial theorem, valid for any real exponent, in 1664-5, inspired by the work of John Wallis's Arithmetic Infinitorum and his method of interpolation. A logarithmic version of the theorem for fractional exponents was discovered independently by James Gregory who wrote down his formula in 1670.