Binomial theorem

Template:Refimprove Template:Image frame 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 Template:Math into a sum involving terms of the form Template:Math, where the exponents Template:Mvar and Template:Mvar are nonnegative integers with Template:Math, and the coefficient Template:Mvar of each term is a specific positive integer depending on Template:Mvar and Template:Mvar. For example (for Template:Math),

$(x+y)^{4}=x^{4}+4x^{3}y+6x^{2}y^{2}+4xy^{3}+y^{4}.$

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

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 Template:Math. There is evidence that the binomial theorem for cubes was known by the 6th century AD in India.

Binomial coefficients, as combinatorial quantities expressing the number of ways of selecting Template:Mvar objects out of Template:Mvar 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.Template:Rp The commentator Halayudha from the 10th century AD explains this method using what is now known as Pascal's triangle. By the 6th century AD, the Indian mathematicians probably knew how to express this as a quotient ${\frac {n!}{(n-k)!k!}}$, and a clear statement of this rule can be found in the 12th century text Lilavati by Bhaskara.

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". Al-Karaji described the triangular pattern of the binomial coefficients and also provided a mathematical proof of both the binomial theorem and Pascal's triangle, using an early form of mathematical induction. 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.Template:Rp

In 1544, Michael Stifel introduced the term "binomial coefficient" and showed how to use them to express $(1+a)^{n}$ in terms of $(1+a)^{n-1}$, via "Pascal's triangle". Blaise Pascal studied the eponymous triangle comprehensively in the treatise Traité du triangle arithmétique (1665). However, the pattern of numbers was already known to the European mathematicians of the late Renaissance, including Stifel, Niccolò Fontana Tartaglia, and Simon Stevin.

Isaac Newton is generally credited with the generalized binomial theorem, valid for any rational exponent.

Theorem statement

According to the theorem, it is possible to expand any power of Template:Math into a sum of the form

$(x+y)^{n}={n \choose 0}x^{n}y^{0}+{n \choose 1}x^{n-1}y^{1}+{n \choose 2}x^{n-2}y^{2}+\cdots +{n \choose n-1}x^{1}y^{n-1}+{n \choose n}x^{0}y^{n},$

where each ${\tbinom {n}{k}}$ is a specific 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 side written as ${\binom {n}{0}}x^{n}+\ldots$.) This formula is also referred to as the binomial formula or the binomial identity. Using summation notation, it can be written as

$(x+y)^{n}=\sum _{k=0}^{n}{n \choose k}x^{n-k}y^{k}=\sum _{k=0}^{n}{n \choose k}x^{k}y^{n-k}.$

The final expression follows from the previous one by the symmetry of Template:Mvar and Template:Mvar 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 Template:Math for Template:Mvar, so that it involves only a single variable. In this form, the formula reads

$(1+x)^{n}={n \choose 0}x^{0}+{n \choose 1}x^{1}+{n \choose 2}x^{2}+\cdots +{n \choose {n-1}}x^{n-1}+{n \choose n}x^{n},$

or equivalently

$(1+x)^{n}=\sum _{k=0}^{n}{n \choose k}x^{k}.$

Examples

The most basic example of the binomial theorem is the formula for the square of Template:Math:

$(x+y)^{2}=x^{2}+2xy+y^{2}.$

The binomial coefficients 1, 2, 1 appearing in this expansion correspond to the second row of Pascal's triangle. (The top "1" of the triangle is considered to be row 0, by convention.) The coefficients of higher powers of Template:Math correspond to lower rows of the triangle:

${\begin{aligned}(x+y)^{3}&=x^{3}+3x^{2}y+3xy^{2}+y^{3},\\[8pt](x+y)^{4}&=x^{4}+4x^{3}y+6x^{2}y^{2}+4xy^{3}+y^{4},\\[8pt](x+y)^{5}&=x^{5}+5x^{4}y+10x^{3}y^{2}+10x^{2}y^{3}+5xy^{4}+y^{5},\\[8pt](x+y)^{6}&=x^{6}+6x^{5}y+15x^{4}y^{2}+20x^{3}y^{3}+15x^{2}y^{4}+6xy^{5}+y^{6},\\[8pt](x+y)^{7}&=x^{7}+7x^{6}y+21x^{5}y^{2}+35x^{4}y^{3}+35x^{3}y^{4}+21x^{2}y^{5}+7xy^{6}+y^{7}.\end{aligned}}$

Several patterns can be observed from these examples. In general, for the expansion Template:Math:

1. the powers of Template:Mvar start at Template:Mvar and decrease by 1 in each term until they reach 0 (with Template:Math, often unwritten);
2. the powers of Template:Mvar start at 0 and increase by 1 until they reach Template:Mvar;
3. the Template:Mvarth row of Pascal's Triangle will be the coefficients of the expanded binomial when the terms are arranged in this way;
4. the number of terms in the expansion before like terms are combined is the sum of the coefficients and is equal to Template:Math; and
5. there will be Template:Math terms in the expression after combining like terms in the expansion.

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

${\begin{aligned}(x+2)^{3}&=x^{3}+3x^{2}(2)+3x(2)^{2}+2^{3}\\&=x^{3}+6x^{2}+12x+8.\end{aligned}}$

For a binomial involving subtraction, the theorem can be applied by using the form Template:Math. This has the effect of changing the sign of every other term in the expansion:

$(x-y)^{3}=(x+(-y))^{3}=x^{3}+3x^{2}(-y)+3x(-y)^{2}+(-y)^{3}=x^{3}-3x^{2}y+3xy^{2}-y^{3}.$

Geometric explanation

Visualisation of binomial expansion up to the 4th power

For positive values of Template:Mvar and Template:Mvar, the binomial theorem with Template:Math is the geometrically evident fact that a square of side Template:Math can be cut into a square of side Template:Mvar, a square of side Template:Mvar, and two rectangles with sides Template:Mvar and Template:Mvar. With Template:Math, the theorem states that a cube of side Template:Math can be cut into a cube of side Template:Mvar, a cube of side Template:Mvar, three Template:Math rectangular boxes, and three Template:Math rectangular boxes.

In calculus, this picture also gives a geometric proof of the derivative $(x^{n})'=nx^{n-1}:$ if one sets $a=x$ and $b=\Delta x,$ interpreting Template:Mvar as an infinitesimal change in Template:Mvar, then this picture shows the infinitesimal change in the volume of an Template:Mvar-dimensional hypercube, $(x+\Delta x)^{n},$ where the coefficient of the linear term (in $\Delta x$) is $nx^{n-1},$ the area of the Template:Mvar faces, each of dimension Template:Math:

$(x+\Delta x)^{n}=x^{n}+nx^{n-1}\Delta x+{\binom {n}{2}}x^{n-2}(\Delta x)^{2}+\cdots .$

Substituting this into the definition of the derivative via a difference quotient and taking limits means that the higher order terms, $(\Delta x)^{2}$ and higher, become negligible, and yields the formula $(x^{n})'=nx^{n-1},$ interpreted as

"the infinitesimal rate of change in volume of an Template:Mvar-cube as side length varies is the area of Template:Mvar of its Template:Math-dimensional faces".

If one integrates this picture, which corresponds to applying the fundamental theorem of calculus, one obtains Cavalieri's quadrature formula, the integral $\textstyle {\int x^{n-1}\,dx={\tfrac {1}{n}}x^{n}}$ – see proof of Cavalieri's quadrature formula for details.

Binomial coefficients

Template:Main The coefficients that appear in the binomial expansion are called binomial coefficients. These are usually written ${\tbinom {n}{k}},$ and pronounced "Template:Mvar choose Template:Mvar".

Formulae

The coefficient of Template:Math is given by the formula

${\binom {n}{k}}={\frac {n!}{k!(n-k)!}},$

which is defined in terms of the factorial function Template:Math. Equivalently, this formula can be written

${\binom {n}{k}}={\frac {n(n-1)\cdots (n-k+1)}{k(k-1)\cdots 1}}=\prod _{\ell =1}^{k}{\frac {n-\ell +1}{\ell }}=\prod _{\ell =0}^{k-1}{\frac {n-\ell }{k-\ell }}$

with Template:Mvar factors in both the numerator and denominator of the fraction. Although this formula involves a fraction, the binomial coefficient ${\tbinom {n}{k}}$ is actually an integer.

Combinatorial interpretation

The binomial coefficient ${\tbinom {n}{k}}$ can be interpreted as the number of ways to choose Template:Mvar elements from an Template:Mvar-element set. This is related to binomials for the following reason: if we write Template:Math as a product

$(x+y)(x+y)(x+y)\cdots (x+y),$

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

Proofs

Combinatorial proof

Example

The coefficient of Template:Math in

${\begin{aligned}(x+y)^{3}&=(x+y)(x+y)(x+y)\\&=xxx+xxy+xyx+{\underline {xyy}}+yxx+{\underline {yxy}}+{\underline {yyx}}+yyy\\&=x^{3}+3x^{2}y+{\underline {3xy^{2}}}+y^{3}\end{aligned}}$

equals ${\tbinom {3}{2}}=3$ because there are three Template:Math strings of length 3 with exactly two Template:Mvars, namely,

$xyy,\;yxy,\;yyx,$

corresponding to the three 2-element subsets of Template:Math, namely,

$\{2,3\},\;\{1,3\},\;\{1,2\},$

where each subset specifies the positions of the Template:Mvar in a corresponding string.

General case

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

• the number of copies of Template:Math in the expansion
• the number of Template:Mvar-character Template:Math strings having Template:Mvar in exactly Template:Mvar positions
• the number of Template:Mvar-element subsets of Template:Math
• ${\tbinom {n}{k}},$ either by definition, or by a short combinatorial argument if one is defining ${\tbinom {n}{k}}$ as ${\tfrac {n!}{k!(n-k)!}}.$

This proves the binomial theorem.

Inductive proof

Induction yields another proof of the binomial theorem. When Template:Math, both sides equal Template:Math, since Template:Math and ${\tbinom {0}{0}}=1.$ Now suppose that the equality holds for a given Template:Mvar; we will prove it for Template:Math. For Template:Math, let Template:Math denote the coefficient of Template:Math in the polynomial Template:Math. By the inductive hypothesis, Template:Math is a polynomial in Template:Mvar and Template:Mvar such that Template:Math is ${\tbinom {n}{k}}$ if Template:Math, and Template:Mvar otherwise. The identity

$(x+y)^{n+1}=x(x+y)^{n}+y(x+y)^{n}$

shows that Template:Math is also a polynomial in Template:Mvar and Template:Mvar, and

$[(x+y)^{n+1}]_{j,k}=[(x+y)^{n}]_{j-1,k}+[(x+y)^{n}]_{j,k-1},$

since if Template:Math, then Template:Math and Template:Math. Now, the right hand side is

${\binom {n}{k}}+{\binom {n}{k-1}}={\binom {n+1}{k}},$

by Pascal's identity. On the other hand, if Template:Math, then Template:Math and Template:Math, so we get Template:Math. Thus

$(x+y)^{n+1}=\sum _{k=0}^{n+1}{\binom {n+1}{k}}x^{n+1-k}y^{k},$

which is the inductive hypothesis with Template:Math substituted for Template:Mvar and so completes the inductive step.

Generalizations

Newton's generalized binomial theorem

Template:Main 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 Template:Mvar, one can define

${r \choose k}={\frac {r(r-1)\cdots (r-k+1)}{k!}}={\frac {(r)_{k}}{k!}},$

where $(\cdot )_{k}$ is the Pochhammer symbol, here standing for a falling factorial. This agrees with the usual definitions when Template:Mvar is a nonnegative integer. Then, if Template:Mvar and Template:Mvar are real numbers with Template:Math, and Template:Mvar is any complex number, one has

${\begin{aligned}(x+y)^{r}&=\sum _{k=0}^{\infty }{r \choose k}x^{r-k}y^{k}\\&=x^{r}+rx^{r-1}y+{\frac {r(r-1)}{2!}}x^{r-2}y^{2}+{\frac {r(r-1)(r-2)}{3!}}x^{r-3}y^{3}+\cdots .\end{aligned}}$

When Template:Mvar is a nonnegative integer, the binomial coefficients for Template:Math are zero, so this equation reduces to the usual binomial theorem, and there are at most Template:Math nonzero terms. For other values of Template:Mvar, the series typically has infinitely many nonzero terms.

For example, Template:Math gives the following series for the square root:

${\sqrt {1+x}}=1+{\frac {1}{2}}x-{\frac {1}{8}}x^{2}+{\frac {1}{16}}x^{3}-{\frac {5}{128}}x^{4}+{\frac {7}{256}}x^{5}-\cdots$

Taking Template:Math, the generalized binomial series gives the geometric series formula, valid for Template:Math:

$(1+x)^{-1}={\frac {1}{1+x}}=1-x+x^{2}-x^{3}+x^{4}-x^{5}+\cdots$

More generally, with Template:Math:

${\frac {1}{(1-x)^{s}}}=\sum _{k=0}^{\infty }{s+k-1 \choose k}x^{k}.$

So, for instance, when Template:Math,

${\frac {1}{\sqrt {1+x}}}=1-{\frac {1}{2}}x+{\frac {3}{8}}x^{2}-{\frac {5}{16}}x^{3}+{\frac {35}{128}}x^{4}-{\frac {63}{256}}x^{5}+\cdots$

Further generalizations

The generalized binomial theorem can be extended to the case where Template:Mvar and Template:Mvar are complex numbers. For this version, one should again assume Template:Math and define the powers of Template:Math and Template:Mvar using a holomorphic branch of log defined on an open disk of radius Template:Math centered at Template:Mvar. The generalized binomial theorem is valid also for elements Template:Mvar and Template:Mvar of a Banach algebra as long as Template:Math, and Template:Mvar is invertible, and Template:Math.

A version of the binomial theorem is valid for the following Pochhammer symbol-like family of polynomials: for a given real constant Template:Mvar, define $x^{(0)}=1$ and

$x^{(n)}=\prod _{k=1}^{n}[x+(k-1)c]$

for $n>0.$ Then

$(a+b)^{(n)}=\sum _{k=0}^{n}{\binom {n}{k}}a^{(n-k)}b^{(k)}.$

The case Template:Math recovers the usual binomial theorem.

More generally, a sequence $\{p_{n}\}_{n=0}^{\infty }$ of polynomials is said to be binomial if

• $\deg p_{n}=n$ for all $n$,
• $p_{0}(0)=1$, and
• $p_{n}(x+y)=\sum _{k=0}^{n}{\binom {n}{k}}p_{k}(x)p_{n-k}(y)$ for all $x$, $y$, and $n$.

An operator $Q$ on the space of polynomials is said to be the basis operator of the sequence $\{p_{n}\}_{n=0}^{\infty }$ if $Qp_{0}=0$ and $Qp_{n}=np_{n-1}$ for all $n\geqslant 1$. A sequence $\{p_{n}\}_{n=0}^{\infty }$ is binomial if and only if its basis operator is a Delta operator. Writing $E^{a}$ for the shift by $a$ operator, the Delta operators corresponding to the above "Pochhammer" families of polynomials are the backward difference $I-E^{-c}$ for $c>0$, the ordinary derivative for $c=0$, and the forward difference $E^{-c}-I$ for $c<0$.

Multinomial theorem

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

$(x_{1}+x_{2}+\cdots +x_{m})^{n}=\sum _{k_{1}+k_{2}+\cdots +k_{m}=n}{\binom {n}{k_{1},k_{2},\ldots ,k_{m}}}x_{1}^{k_{1}}x_{2}^{k_{2}}\cdots x_{m}^{k_{m}},$

where the summation is taken over all sequences of nonnegative integer indices Template:Math through Template:Math such that the sum of all Template:Math is Template:Mvar. (For each term in the expansion, the exponents must add up to Template:Mvar). The coefficients ${\tbinom {n}{k_{1},\cdots ,k_{m}}}$ are known as multinomial coefficients, and can be computed by the formula

${\binom {n}{k_{1},k_{2},\ldots ,k_{m}}}={\frac {n!}{k_{1}!\cdot k_{2}!\cdots k_{m}!}}.$

Combinatorially, the multinomial coefficient ${\tbinom {n}{k_{1},\cdots ,k_{m}}}$ counts the number of different ways to partition an Template:Mvar-element set into disjoint subsets of sizes Template:Math.

Template:Anchor 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

$(x_{1}+y_{1})^{n_{1}}\dotsm (x_{d}+y_{d})^{n_{d}}=\sum _{k_{1}=0}^{n_{1}}\dotsm \sum _{k_{d}=0}^{n_{d}}{\binom {n_{1}}{k_{1}}}x_{1}^{k_{1}}y_{1}^{n_{1}-k_{1}}\dotsc {\binom {n_{d}}{k_{d}}}x_{d}^{k_{d}}y_{d}^{n_{d}-k_{d}}.$

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

$(x+y)^{\alpha }=\sum _{\nu \leq \alpha }{\binom {\alpha }{\nu }}x^{\nu }y^{\alpha -\nu }.$

General Leibniz rule

The general Leibniz rule gives the Template:Mvarth derivative of a product of two functions in a form similar to that of the binomial theorem:

$(fg)^{(n)}(x)=\sum _{k=0}^{n}{\binom {n}{k}}f^{(n-k)}(x)g^{(k)}(x).$

Here, the superscript Template:Math indicates the Template:Mvarth derivative of a function. If one sets Template:Math and Template:Math, and then cancels the common factor of Template:Math from both sides of the result, the ordinary binomial theorem is recovered.

Applications

Multiple-angle identities

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,

$\cos \left(nx\right)+i\sin \left(nx\right)=\left(\cos x+i\sin x\right)^{n}.$

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 Template:Math and Template:Math. For example, since

$\left(\cos x+i\sin x\right)^{2}=\cos ^{2}x+2i\cos x\sin x-\sin ^{2}x,$

De Moivre's formula tells us that

$\cos(2x)=\cos ^{2}x-\sin ^{2}x\quad {\text{and}}\quad \sin(2x)=2\cos x\sin x,$

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

$\left(\cos x+i\sin x\right)^{3}=\cos ^{3}x+3i\cos ^{2}x\sin x-3\cos x\sin ^{2}x-i\sin ^{3}x,$

De Moivre's formula yields

$\cos(3x)=\cos ^{3}x-3\cos x\sin ^{2}x\quad {\text{and}}\quad \sin(3x)=3\cos ^{2}x\sin x-\sin ^{3}x.$

In general,

$\cos(nx)=\sum _{k{\text{ even}}}(-1)^{k/2}{n \choose k}\cos ^{n-k}x\sin ^{k}x$

and

$\sin(nx)=\sum _{k{\text{ odd}}}(-1)^{(k-1)/2}{n \choose k}\cos ^{n-k}x\sin ^{k}x.$

Series for e

The [[e (mathematical constant)|number Template:Mvar]] is often defined by the formula

$e=\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}.$

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

$\left(1+{\frac {1}{n}}\right)^{n}=1+{n \choose 1}{\frac {1}{n}}+{n \choose 2}{\frac {1}{n^{2}}}+{n \choose 3}{\frac {1}{n^{3}}}+\cdots +{n \choose n}{\frac {1}{n^{n}}}.$

The Template:Mvarth term of this sum is

${n \choose k}{\frac {1}{n^{k}}}={\frac {1}{k!}}\cdot {\frac {n(n-1)(n-2)\cdots (n-k+1)}{n^{k}}}$

As Template:Math, the rational expression on the right approaches Template:Math, and therefore

$\lim _{n\to \infty }{n \choose k}{\frac {1}{n^{k}}}={\frac {1}{k!}}.$

This indicates that Template:Mvar can be written as a series:

$e=\sum _{k=0}^{\infty }{\frac {1}{k!}}={\frac {1}{0!}}+{\frac {1}{1!}}+{\frac {1}{2!}}+{\frac {1}{3!}}+\cdots .$

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

Probability

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 $\{X_{t}\}_{t\in S}$ with probability of success $p\in [0,1]$ all not happening is

$P\left(\bigcap _{t\in S}X_{t}^{C}\right)=(1-p)^{|S|}=\sum _{n=0}^{|S|}{|S| \choose n}(-p)^{n}.$

A useful upper bound for this quantity is $e^{-pn}.$

The binomial theorem in abstract algebra

The binomial theorem is valid more generally for any elements Template:Mvar and Template:Mvar of a semiring satisfying Template:Math. The theorem is true even more generally: alternativity suffices in place of associativity.

The binomial theorem can be stated by saying that the polynomial sequence Template:Math is of binomial type.

In popular culture

• 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."
• 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.

See also

• Binomial approximation
• Binomial distribution
• Binomial inverse theorem
• Stirling's approximation

Further reading

• Template:Cite journal
• Template:Cite book

External links

• Template:SpringerEOM
• Binomial Theorem by Stephen Wolfram, and "Binomial Theorem (Step-by-Step)" by Bruce Colletti and Jeff Bryant, Wolfram Demonstrations Project, 2007.