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In , the binomial theorem describes the algebraic expansion of
of a . Hence it is referred to as binomial expansion. According to the theorem, it is possible to expand the power (x + y)n into a
involving terms of the form axbyc, where the exponents b and c are
with b + c = n, and the
a of each term is a specific
depending on n and b. When an exponent is zero, the corresponding power expression is usually omitted from the term. For example,
The coefficient a in the term of axbyc is known as the
(the two have the same value). These coefficients for varying n and b can be arranged to form . These numbers also arise in , where
gives the number of different
that can be chosen from an n-element .
This formula and the triangular arrangement of the binomial coefficients are often attributed to , who described them in the 17th century, but they were known to many mathematicians who preceded him. For instance, the 4th century B.C.
mentioned the special case of the binomial theorem for exponent 2 as did the 3rd century B.C.
to higher orders. A more general binomial theorem and the so-called "" were known in the 10th century A.D. to Indian mathematician .
, in the 11th century was aware of a more general binomial theorem, along with Persian poet and mathematician , and in the 13th century to
, who all derived similar results. Al-Karaji also provided a
of both the binomial theorem and Pascal's triangle, using a primitive form of .
is generally credited with the generalised binomial theorem, valid for any rational exponent.
According to the theorem, it is possible to expand any power of x + y into a sum of the form
where each
is a specific positive integer known as a . This formula is also referred to as the binomial formula or the binomial identity. Using , 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
1 for y, so that it involves only a single . In this form, the formula reads
or equivalently
Pascal's triangle
The most basic example of the binomial theorem is the formula for the
The binomial coefficients 1, 2, 1 appearing in this expansion correspond to the second row of Pascal's triangle (Note that the top is row 0). The coefficients of higher powers of x + y correspond to later rows of the triangle:
Notice that
the powers of x go down until it reaches 0 (), starting value is n (the n in .)
the powers of y go up from 0 () until it reaches n (also the n in .)
the nth row of the Pascal's Triangle will be the coefficients of the expanded binomial.
for each line, the number of products (i.e. the sum of the coefficients) is equal to .
for each line, the number of product groups is equal to .
The binomial theorem can be applied to the powers of any binomial. For example,
For a binomial involving subtraction, the theorem can be applied as long as the
of the second term is used. This has the effect of changing the sign of every other term in the expansion:
Another useful example is that of the expansion of the following square roots:
Sometimes it may be useful to expand negative exponents when :
Visualisation of binomial expansion up to the 4th power
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 , this picture also gives a geometric proof of the
if one sets
interpreting b as an infinitesimal change in a, then this picture shows the infinitesimal change in the volume of an n-dimensional ,
where the coefficient of the linear term (in ) is
the area of the n faces, each of dimension
Substituting this into the
and taking limits means that the higher order terms –
and higher – become negligible, and yields the formula
interpreted as
"the infinitesimal change in volume of an n-cube as side length varies is the area of n of its -dimensional faces".
If one integrates this picture, which corresponds to applying the , one obtains , the integral
for details.
Main article:
The coefficients that appear in the binomial expansion are called binomial coefficients. These are usually written , and pronounced “n choose k”.
The coefficient of xn-kyk is given by the formula
which is defined in terms of the
function n!. Equivalently, this formula can be written
with k factors in both the numerator and denominator of the . Note that, although this formula involves a fraction, the binomial coefficient
is actually an .
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
then, according to the , 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 xn, corresponding to choosing x from each binomial. However, there will be several terms of the form xn-2y2, one for each way of choosing exactly two binomials to contribute a y. Therefore, after , the coefficient of xn-2y2 will be equal to the number of ways to choose exactly 2 elements from an n-element set.
The coefficient of xy2 in
because there are three x,y strings of length 3 with exactly two y's, 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 e1e2 ... e n where each e i is x or y. Rearranging factors shows that each product equals xn-kyk for some k between 0 and n. For a given k, the following are proved equal in succession:
the number of copies of xn - kyk 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}
(this is either by definition, or by a short combinatorial argument if one is defining
This proves the binomial theorem.
yields another proof of the binomial theorem (1). When n = 0, both sides equal 1, since x0 = 1 and . Now suppose that (1) holds for a given n; we will prove it for n + 1. For j, k ≥ 0, let [?(x, y)] j,k denote the coefficient of xjyk in the polynomial ?(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 also is 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 . 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.
Main article:
Around 1665,
generalised the formula to allow real exponents other than nonnegative integers. In addition, the formula can be generalised to complex exponents. In this generalisation, the finite sum is replaced by an . In order to do this, one needs to give meaning to binomial coefficients with an arbitrary upper index, which cannot be done using the above for however factoring out (n - k)! from numerator and denominator in that formula, and replacing n by r which now stands for an arbitrary number, one can define
here standing for a . Then, if x and y are real numbers with |x| & |y|, and r is any , one has
When r is a nonnegative integer, the binomial coefficients for k & r are zero, so (2) specializes to (1), and there are at most r + 1 nonzero terms. For other values of r, the series (2) has infinitely many nonzero terms, at least if x and y are nonzero.
This is important when one is working with infinite series and would like to represent them in terms of .
Taking r = -s leads to a useful formula:
Further specializing to s = 1 yields the .
Formula (2) can be generalised to the case where x and y are . For this version, one should assume |x| & |y| and define the powers of x + y and x using a
defined on an open disk of radius |x| centered at x.
Formula (2) is valid also for elements x and y of a
as long as xy = yx, x is invertible, and ||y/x|| & 1.
Main article:
The binomial theorem can be generalised 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 k1 through km such that the sum of all ki 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
an n-element set into
of sizes k1, ..., km.
The multi-binomial theorem[]
It is often useful when working in more dimensions, to deal with products of binomial expressions. By the binomial theorem this is equal to
This may be written more concisely, by , as
the binomial theorem can be combined with
and . 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,
is often defined by the formula
Applying the binomial theorem to this expression yields the usual
for e. In particular:
The kth term of this sum is
As n → ∞, the rational expression on the right approaches one, and therefore
This indicates that e can be written as a series:
Indeed, since each term of the binomial expansion is an
of n, it follows from the
for series that the sum of this infinite series is equal to e.
In finding the derivative of the power function, f(x) = xn, by using the definition of derivative, the expansion of (x + h)n is employed.
To indicate the formula for the derivative of order n of the product of two functions, the formula of the binomial theorem is used symbolically.
Formula (1) is valid more generally for any elements x and y of a
satisfying xy = yx. The
is true even more generally:
suffices in place of .
The binomial theorem can be stated by saying that the
{ 1, x, x2, x3, ... } is of .
The binomial theorem is mentioned in the
in the comic opera .
is described by Sherlock Holmes as having written .
, The American Mathematical Monthly 56:3 (1949), pp. 147–157
Sandler, Stanley (2011). An Introduction to Applied Statistical Thermodynamics. Hoboken NJ: John Wiley & Sons, Inc.  .
Landau, James A ().
(MAILING LIST EMAIL). Archives of Historia Matematica.
Bourbaki: History of mathematics
- inductive proofs
This is to guarantee convergence. Depending on r, the series may also converge sometimes when |x| = |y|.
Seely, Robert T. (1973). Calculus of One and Several Variables. Glenview: Scott, Foresman.  .
Bag, Amulya Kumar (1966). "Binomial theorem in ancient India". Indian J. History Sci 1 (1): 68–74.
Barth, Nils R. (November 2004). "Computing Cavalieri's Quadrature Formula by a Symmetry of the n-Cube". The American Mathematical Monthly (Mathematical Association of America) 111 (9): 811–813. :.  .  , ,
Graham, R Knuth, D Patashnik, Oren (1994). "(5) Binomial Coefficients". Concrete Mathematics (2nd ed.). Addison Wesley. pp. 153–256.  .  .
The Wikibook
has a page on the topic of:
Solomentsev, E.D. (2001), , in Hazewinkel, Michiel, , ,  
by Bruce Colletti and Jeff Bryant, , 2007.
This article incorporates material from
on , which is licensed under the .
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