Binomial theorem

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The binomial coefficient $(\genfrac{}{}{0ex}{}{n}{k})$ appears as the Template:Mvarth entry in the Template:Mvarth row of Pascal's triangle (where the top is the 0th row $(\genfrac{}{}{0ex}{}{0}{0})$). Each entry is the sum of the two above it.

In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, the power Template:Tmath expands into a polynomial with terms of the form Template:Tmath, where the exponents Template:Tmath and Template:Tmath are nonnegative integers satisfying Template:Tmath and the coefficient Template:Tmath of each term is a specific positive integer depending on Template:Tmath and Template:Tmath. For example, for Template:Tmath, $$(x+y{)}^4=x^4+4x^{3}y+6x^2{y}^2+4x{y}^3+y^4.$$

The coefficient Template:Tmath in each term Template:Tmath is known as the binomial coefficient Template:Tmath or Template:Tmath (the two have the same value). These coefficients for varying Template:Tmath and Template:Tmath can be arranged to form Pascal's triangle. These numbers also occur in combinatorics, where Template:Tmath gives the number of different combinations (i.e. subsets) of Template:Tmath elements that can be chosen from an Template:Tmath-element set. Therefore Template:Tmath is usually pronounced as "Template:Tmath choose Template:Tmath".

Statement

According to the theorem, the expansion of any nonnegative integer power Template:Mvar of the binomial x + y is a sum of the form $$(x+y{)}^n=(\genfrac{}{}{0ex}{}{n}{0})x^n{y}^0+(\genfrac{}{}{0ex}{}{n}{1})x^{n-1}{y}^1+(\genfrac{}{}{0ex}{}{n}{2})x^{n-2}{y}^2+\cdots +(\genfrac{}{}{0ex}{}{n}{n})x^0{y}^n,$$ where each $(\genfrac{}{}{0ex}{}{n}{k})$ is a positive integer known as a binomial coefficient, defined as

$$(\genfrac{}{}{0ex}{}{n}{k})=\frac{n!}{k!(n-k)!}=\frac{n(n-1)(n-2)\cdots (n-k+1)}{k(k-1)(k-2)\cdots 2\cdot 1}.$$

This formula is also referred to as the binomial formula or the binomial identity. Using summation notation, it can be written more concisely as $$(x+y{)}^n=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})x^{n-k}{y}^k=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})x^k{y}^{n-k}.$$

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 symmetric, $(\genfrac{}{}{0ex}{}{n}{k})=(\genfrac{}{}{0ex}{}{n}{n-k}).$

A simple variant of the binomial formula is obtained by substituting 1 for Template:Mvar, so that it involves only a single variable. In this form, the formula reads $$\begin{array}{rl}(x+1{)}^n& =(\genfrac{}{}{0ex}{}{n}{0})x^0+(\genfrac{}{}{0ex}{}{n}{1})x^1+(\genfrac{}{}{0ex}{}{n}{2})x^2+\cdots +(\genfrac{}{}{0ex}{}{n}{n})x^n\\ & =\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})x^k.\phantom{)}\end{array}$$

Examples

The first few cases of the binomial theorem are: $$\begin{array}{rl}(x+y{)}^0& =1,\\ (x+y{)}^1& =x+y,\\ (x+y{)}^2& =x^2+2xy+y^2,\\ (x+y{)}^3& =x^3+3x^{2}y+3x{y}^2+y^3,\\ (x+y{)}^4& =x^4+4x^{3}y+6x^2{y}^2+4x{y}^3+y^4,\end{array}$$ In general, for the expansion of (x + y)ⁿ on the right side in the Template:Mvarth row (numbered so that the top row is the 0th row):

• the exponents of Template:Mvar in the terms are n, n − 1, ..., 2, 1, 0 (the last term implicitly contains x⁰ = 1);
• the exponents of Template:Mvar in the terms are 0, 1, 2, ..., n − 1, n (the first term implicitly contains y⁰ = 1);
• the coefficients form the Template:Mvarth row of Pascal's triangle;
• before combining like terms, there are 2ⁿ terms xⁱy^j in the expansion (not shown);
• after combining like terms, there are n + 1 terms, and their coefficients sum to 2ⁿ.

An example illustrating the last two points: $$\begin{array}{rlr}(x+y{)}^3& =xxx+xxy+xyx+xyy+yxx+yxy+yyx+yyy& (2^3\text{ terms})\\ & =x^3+3x^{2}y+3x{y}^2+y^3& (3+1\text{ terms})\end{array}$$ with $1+3+3+1=2^3$.

A simple example with a specific positive value of y: $$\begin{array}{rl}(x+2{)}^3& =x^3+3x^2(2)+3x(2{)}^2+2^3\\ & =x^3+6x^2+12x+8.\end{array}$$

A simple example with a specific negative value of y: $$\begin{array}{rl}(x-2{)}^3& =x^3-3x^2(2)+3x(2{)}^2-2^3\\ & =x^3-6x^2+12x-8.\end{array}$$

Geometric explanation

File:Binomial theorem visualisation.svg

Visualisation of binomial expansion up to the 4th power

For positive values of Template:Mvar and Template:Mvar, 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 Template:Mvar, a square of side Template:Mvar, and two rectangles with sides Template:Mvar and Template:Mvar. With n = 3, the theorem states that a cube of side a + b can be cut into a cube of side Template:Mvar, a cube of side Template:Mvar, 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 $(x^n{)}^{\prime }=n{x}^{n-1}:$ if one sets $a=x$ and $b=\mathrm{\Delta }x,$ 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, $(x+\mathrm{\Delta }x{)}^n,$ where the coefficient of the linear term (in $\mathrm{\Delta }x$) is $n{x}^{n-1},$ the area of the Template:Mvar faces, each of dimension n − 1: $$(x+\mathrm{\Delta }x{)}^n=x^n+n{x}^{n-1}\mathrm{\Delta }x+(\genfrac{}{}{0ex}{}{n}{2})x^{n-2}(\mathrm{\Delta }x{)}^2+\cdots .$$ Substituting this into the definition of the derivative via a difference quotient and taking limits means that the higher order terms, $(\mathrm{\Delta }x{)}^2$ and higher, become negligible, and yields the formula $(x^n{)}^{\prime }=n{x}^{n-1},$ 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 (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 $\int x^{n-1}dx=\frac{1}{n}{x}^n$ – 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 $(\genfrac{}{}{0ex}{}{n}{k}),$ and pronounced "Template:Mvar choose Template:Mvar".

Formulas

The coefficient of x^n−ky^k is given by the formula $$(\genfrac{}{}{0ex}{}{n}{k})=\frac{n!}{k!(n-k)!},$$ which is defined in terms of the factorial function n!. Equivalently, this formula can be written $$(\genfrac{}{}{0ex}{}{n}{k})=\frac{n(n-1)\cdots (n-k+1)}{k(k-1)\cdots 1}=\prod _{\ell =1}^k\frac{n-\ell +1}{\ell }=\prod _{\ell =0}^{k-1}\frac{n-\ell }{k-\ell }$$ with Template:Mvar factors in both the numerator and denominator of the fraction. Although this formula involves a fraction, the binomial coefficient $(\genfrac{}{}{0ex}{}{n}{k})$ is actually an integer.

Combinatorial interpretation

The binomial coefficient $(\genfrac{}{}{0ex}{}{n}{k})$ can be interpreted as the number of ways to choose Template:Mvar elements from an Template:Mvar-element set (a combination). This is related to binomials for the following reason: if we write (x + y)ⁿ 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 xⁿ, corresponding to choosing Template:Mvar 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 Template:Mvar. Therefore, after combining like terms, the coefficient of xⁿ⁻²y² will be equal to the number of ways to choose exactly 2 elements from an Template:Mvar-element set.

Proofs

Combinatorial proof

Expanding (x + y)ⁿ yields the sum of the 2ⁿ products of the form e₁e₂ ... e_n where each e_i is Template:Mvar or Template:Mvar. Rearranging factors shows that each product equals x^n−ky^k for some Template:Mvar between 0 and Template:Mvar. For a given Template:Mvar, the following are proved equal in succession:

• the number of terms equal to x^n−ky^k in the expansion
• the number of Template:Mvar-character x,y strings having Template:Mvar in exactly Template:Mvar positions
• the number of Template:Mvar-element subsets of Template:Mset
• $(\genfrac{}{}{0ex}{}{n}{k}),$ either by definition, or by a short combinatorial argument if one is defining $(\genfrac{}{}{0ex}{}{n}{k})$ as $\frac{n!}{k!(n-k)!}.$

This proves the binomial theorem.

Example

The coefficient of xy² in $$\begin{array}{rl}(x+y{)}^3& =(x+y)(x+y)(x+y)\\ & =xxx+xxy+xyx+\underset{\_}{xyy}+yxx+\underset{\_}{yxy}+\underset{\_}{yyx}+yyy\\ & =x^3+3x^{2}y+\underset{\_}{3x{y}^2}+y^3\end{array}$$ equals $(\genfrac{}{}{0ex}{}{3}{2})=3$ because there are three x,y strings of length 3 with exactly two Template:Mvar's, namely, $$xyy,yxy,yyx,$$ corresponding to the three 2-element subsets of Template:Mset, namely, $$\{2,3\},\{1,3\},\{1,2\},$$ where each subset specifies the positions of the Template:Mvar in a corresponding string.

Inductive proof

Induction yields another proof of the binomial theorem. When n = 0, both sides equal 1, since x⁰ = 1 and $(\genfrac{}{}{0ex}{}{0}{0})=1.$ Now suppose that the equality holds for a given Template:Mvar; we will prove it for n + 1. For j, k ≥ 0, let [f(x, y)]_j,k denote the coefficient of x^jy^k in the polynomial f(x, y). By the inductive hypothesis, (x + y)ⁿ is a polynomial in Template:Mvar and Template:Mvar such that [(x + y)ⁿ]_j,k is $(\genfrac{}{}{0ex}{}{n}{k})$ if j + k = n, and Template:Mvar otherwise. The identity $$(x+y{)}^{n+1}=x(x+y{)}^n+y(x+y{)}^n$$ shows that (x + y)ⁿ⁺¹ 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 j + k = n + 1, then (j − 1) + k = n and j + (k − 1) = n. Now, the right hand side is $$(\genfrac{}{}{0ex}{}{n}{k})+(\genfrac{}{}{0ex}{}{n}{k-1})=(\genfrac{}{}{0ex}{}{n+1}{k}),$$ by Pascal's identity. 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 $$(x+y{)}^{n+1}=\sum _{k=0}^{n+1}(\genfrac{}{}{0ex}{}{n+1}{k})x^{n+1-k}{y}^k,$$ which is the inductive hypothesis with n + 1 substituted for Template:Mvar and so completes the inductive step.

Generalizations

Generalized binomial theorem

The standard binomial theorem, as discussed above, is concerned with $(x+y{)}^n$ where the exponent $n$ 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 Template:Mvar, one can define $$(\genfrac{}{}{0ex}{}{r}{k})=\frac{r(r-1)\cdots (r-k+1)}{k!}=\frac{r^{\underset{\_}{k}}}{k!},$$ where the last equation introduces modern notation for the 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:Abs > Template:Abs, and Template:Mvar is any complex number, one has $$\begin{array}{rl}(x+y{)}^r& =\sum _{k=0}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{r}{k})x^{r-k}{y}^k\\ & =x^r+r{x}^{r-1}y+\frac{r(r-1)}{2!}{x}^{r-2}{y}^2+\frac{r(r-1)(r-2)}{3!}{x}^{r-3}{y}^3+\cdots .\end{array}$$

When Template:Mvar 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 Template:Mvar, the series has infinitely many nonzero terms.

For example, r = 1/2 gives the following series for the square root: $$\sqrt{1+x}=1+\frac{1}{2}x-\frac{1}{8}{x}^2+\frac{1}{16}{x}^3-\frac{5}{128}{x}^4+\frac{7}{256}{x}^5-\cdots .$$

With r = −1, the generalized binomial series becomes: $$(1+x{)}^{-1}=\frac{1}{1+x}=1-x+x^2-x^3+x^4-x^5+\cdots .$$ which is the geometric series sum formula for the convergent case Template:Abs < 1, whose common ratio is −x.

More generally, with r = −s, we have for Template:Abs < 1: $$\frac{1}{(1+x{)}^s}=\sum _{k=0}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{-s}{k})x^k=\sum _{k=0}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{s+k-1}{k})(-1{)}^k{x}^k.$$

So, for instance, when s = 1/2, $$\frac{1}{\sqrt{1+x}}=1-\frac{1}{2}x+\frac{3}{8}{x}^2-\frac{5}{16}{x}^3+\frac{35}{128}{x}^4-\frac{63}{256}{x}^5+\cdots .$$

Replacing Template:Mvar with Template:Mvar yields: $$\frac{1}{(1-x{)}^s}=\sum _{k=0}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{s+k-1}{k})(-1{)}^k(-x{)}^k=\sum _{k=0}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{s+k-1}{k})x^k.$$

So, for instance, when s = 1/2, we have for Template:Abs < 1: $$\frac{1}{\sqrt{1-x}}=1+\frac{1}{2}x+\frac{3}{8}{x}^2+\frac{5}{16}{x}^3+\frac{35}{128}{x}^4+\frac{63}{256}{x}^5+\cdots .$$

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:Abs > Template:Abs and define the powers of x + y and Template:Mvar using a holomorphic branch of log defined on an open disk of radius Template:Abs 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 xy = yx, and Template:Mvar is invertible, and Template:Norm < 1.

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 $$(a+b{)}^{(n)}=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})a^{(n-k)}{b}^{(k)}.$$ The case c = 0 recovers the usual binomial theorem.

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

• $\mathrm{deg}{p}_n=n$ for all $n$,
• $p_0(0)=1$, and
• $p_n(x+y)=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})p_k(x)p_{n-k}(y)$ for all $x$, $y$, and $n$.

An operator $Q$ on the space of polynomials is said to be the basis operator of the sequence $\{p_n{\}}_{n=0}^{\mathrm{\infty }}$ if $Q{p}_0=0$ and $Q{p}_n=n{p}_{n-1}$ for all $n⩾1$. A sequence $\{p_n{\}}_{n=0}^{\mathrm{\infty }}$ 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

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

$$(x_1+x_2+\cdots +x_m{)}^n=\sum _{k_1+k_2+\cdots +k_m=n}(\genfrac{}{}{0ex}{}{n}{k_1,k_2,\dots ,k_m})x_1^{k_1}{x}_2^{k_2}\cdots x_m^{k_m},$$

where the summation is taken over all sequences of nonnegative integer indices k₁ through k_m such that the sum of all k_i is Template:Mvar. (For each term in the expansion, the exponents must add up to Template:Mvar). The coefficients $(\genfrac{}{}{0ex}{}{n}{k_1,\cdots ,k_m})$ are known as multinomial coefficients, and can be computed by the formula $$(\genfrac{}{}{0ex}{}{n}{k_1,k_2,\dots ,k_m})=\frac{n!}{k_1!\cdot k_2!\cdots k_m!}.$$

Combinatorially, the multinomial coefficient $(\genfrac{}{}{0ex}{}{n}{k_1,\cdots ,k_m})$ counts the number of different ways to partition an Template:Mvar-element set into disjoint subsets of sizes k₁, ..., k_m.

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 $$(x_1+y_1{)}^{n_1}\cdots (x_d+y_d{)}^{n_d}=\sum _{k_1=0}^{n_1}\cdots \sum _{k_d=0}^{n_d}(\genfrac{}{}{0ex}{}{n_1}{k_1})x_1^{k_1}{y}_1^{n_1-k_1}\dots (\genfrac{}{}{0ex}{}{n_d}{k_d})x_d^{k_d}{y}_d^{n_d-k_d}.$$

This may be written more concisely, by multi-index notation, as $$(x+y{)}^{\alpha }=\sum _{\nu \le \alpha }(\genfrac{}{}{0ex}{}{\alpha }{\nu })x^{\nu }{y}^{\alpha -\nu }.$$

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

Here, the superscript (n) indicates the Template:Mvarth derivative of a function, $f^{(n)}(x)=\frac{d^n}{d{x}^n}f(x)$. If one sets f(x) = e^ax and g(x) = e^bx, cancelling the common factor of e^{(a + b)x} 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 $n=2$. Greek mathematician Diophantus cubed various binomials, including $x-1$. Indian mathematician Aryabhata's method for finding cube roots, from around 510 AD, suggests that he knew the binomial formula for exponent $n=3$.

Binomial coefficients, as combinatorial quantities expressing the number of ways of selecting Template:Mvar objects out of Template:Mvar without replacement (combinations), were of interest to ancient Indian mathematicians. The Jain Bhagavati Sutra (c. 300 BC) describes the number of combinations of philosophical categories, senses, or other things, with correct results up through Template:Tmath (probably obtained by listing all possibilities and counting them) and a suggestion that higher combinations could likewise be found. The Chandaḥśāstra by the Indian lyricist Piṅgala (3rd or 2nd century BC) 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" (meru-prastāra) for counting metres is equivalent to Pascal's triangle. Varāhamihira (6th century AD) 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 Template:Tmath, and clear statements of this rule can be found in Śrīdhara's Pāṭīgaṇita (8th–9th century), Mahāvīra's Gaṇita-sāra-saṅgraha (c. 850), and Bhāskara II's Līlāvatī (12th century).Template:RTemplate:R

The Persian mathematician al-Karajī (953–1029) 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 (12th century), there credited to al-Karajī.Template:RTemplate:R 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 (Pascal's triangle turned on its side) up to Template:Tmath and a rule for generating them equivalent to the recurrence relation Template:Tmath.Template:R 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 (13th century). In 1544, Michael Stifel introduced the term "binomial coefficient" and showed how to use them to express $(1+x{)}^n$ in terms of $(1+x{)}^{n-1}$, 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 $n=\frac{1}{2}$, can be found in the work of Henry Briggs' Arithmetica Logarithmica (1624).Template:R 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.Template:R A logarithmic version of the theorem for fractional exponents was discovered independently by James Gregory who wrote down his formula in 1670.

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

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 cos(nx) and sin(nx). For example, since $${(\cos x+i\sin x)}^2={\cos }^{}x+2i\cos x\sin x-{\sin }^{}x=({\cos }^{}x-{\sin }^{}x)+i(2\cos x\sin x),$$ But De Moivre's formula identifies the left side with $(\cos x+i\sin x{)}^2=\cos (2x)+i\sin (2x)$, so $$\cos (2x)={\cos }^{}x-{\sin }^{}x\text{and}\sin (2x)=2\cos x\sin x,$$ which are the usual double-angle identities. Similarly, since $${(\cos x+i\sin x)}^3={\cos }^{}x+3i{\cos }^{}x\sin x-3\cos x{\sin }^{}x-i{\sin }^{}x,$$ De Moivre's formula yields $$\cos (3x)={\cos }^{}x-3\cos x{\sin }^{}x\text{and}\sin (3x)=3{\cos }^{}x\sin x-{\sin }^{}x.$$ In general, $$\cos (nx)=\sum _{k\text{ even}}(-1{)}^{k/2}(\genfrac{}{}{0ex}{}{n}{k}){\cos }^{}x{\sin }^{}x$$ and $$\sin (nx)=\sum _{k\text{ odd}}(-1{)}^{(k-1)/2}(\genfrac{}{}{0ex}{}{n}{k}){\cos }^{}x{\sin }^{}x.$$There are also similar formulas using Chebyshev polynomials.

Series for e

The [[e (mathematical constant)|number Template:Mvar]] is often defined by the formula $$e=\underset{n\to \mathrm{\infty }}{\lim }{(1+\frac{1}{n})}^n.$$

Applying the binomial theorem to this expression yields the usual infinite series for Template:Mvar. In particular: $${(1+\frac{1}{n})}^n=1+(\genfrac{}{}{0ex}{}{n}{1})\frac{1}{n}+(\genfrac{}{}{0ex}{}{n}{2})\frac{1}{n^2}+(\genfrac{}{}{0ex}{}{n}{3})\frac{1}{n^3}+\cdots +(\genfrac{}{}{0ex}{}{n}{n})\frac{1}{n^n}.$$

The Template:Mvarth term of this sum is $$(\genfrac{}{}{0ex}{}{n}{k})\frac{1}{n^k}=\frac{1}{k!}\cdot \frac{n(n-1)(n-2)\cdots (n-k+1)}{n^k}$$

As n → ∞, the rational expression on the right approaches 1, and therefore $$\underset{n\to \mathrm{\infty }}{\lim }(\genfrac{}{}{0ex}{}{n}{k})\frac{1}{n^k}=\frac{1}{k!}.$$

This indicates that Template:Mvar can be written as a series: $$e=\sum _{k=0}^{\mathrm{\infty }}\frac{1}{k!}=\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\cdots .$$

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 $$P\left(\bigcap _{t\in S}{X}_t^C\right)=(1-p{)}^{|S|}=\sum _{n=0}^{|S|}(\genfrac{}{}{0ex}{}{|S|}{n})(-p{)}^n.$$ An upper bound for this quantity is $e^{-p|S|}.$

In abstract algebra

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.

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

See also

• Binomial approximation
• Binomial distribution
• Binomial inverse theorem
• Binomial coefficient
• Stirling's approximation
• Tannery's theorem
• Polynomials calculating sums of powers of arithmetic progressions
• q-binomial theorem

External links

• 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.