binomial theorem

According to the theorem, the expansion of any nonnegative integer power {{mvar|n}} of the binomial {{math|x + y}} is a sum of the form

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

where each $\backslash tbinom\; nk$ is a positive integer known as a binomial coefficient, defined as

$$\backslash binom\; nk\; =\; \backslash frac\{n!\}\{k!\backslash ,(n-k)!\}\; =\; \backslash frac\{n(n-1)(n-2)\backslash cdots(n-k\; +\; 1)\}\{k(k-1)(k-2)\backslash cdots2\backslash cdot1\}.$$

This formula is also referred to as the binomial formula or the binomial identity. Using summation notation, it can be written more concisely as

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

The final expression follows from the previous one by the symmetry of {{mvar|x}} and {{mvar|y}} in the first expression, and by comparison it follows that the sequence of binomial coefficients in the formula is symmetrical, $\backslash binom\; nk\; =\; \backslash binom\; n\{n-k\}.$

A simple variant of the binomial formula is obtained by substituting {{math|1}} for {{mvar|y}}, so that it involves only a single variable. In this form, the formula reads

$$\backslash begin\{align\}$$

(x+1)^n

&= {n \choose 0}x^0 + {n \choose 1}x^1 + {n \choose 2}x^2 + \cdots + {n \choose n}x^n \\[4mu]

&= \sum_{k=0}^n {n \choose k}x^k. \vphantom{\Bigg)}

\end{align}

The first few cases of the binomial theorem are:

$$\backslash begin\{align\}$$

(x+y)^0 & = 1, \\[8pt]

(x+y)^1 & = x + y, \\[8pt]

(x+y)^2 & = x^2 + 2xy + y^2, \\[8pt]

(x+y)^3 & = x^3 + 3x^2y + 3xy^2 + y^3, \\[8pt]

(x+y)^4 & = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4,

\end{align}

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

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

An example illustrating the last two points: $$\backslash begin\{align\}$$

(x+y)^3 & = xxx + xxy + xyx + xyy + yxx + yxy + yyx + yyy & (2^3 \text{ terms}) \\

& = x^3 + 3x^2y + 3xy^2 + y^3 & (3 + 1 \text{ terms})

\end{align} with $1\; +\; 3\; +\; 3\; +\; 1\; =\; 2^3$.

A simple example with a specific positive value of {{math|y}}:

$$\backslash begin\{align\}$$

(x+2)^3 &= x^3 + 3x^2(2) + 3x(2)^2 + 2^3 \\

&= x^3 + 6x^2 + 12x + 8.

\end{align}

A simple example with a specific negative value of {{math|y}}:

$$\backslash begin\{align\}$$

(x-2)^3 &= x^3 - 3x^2(2) + 3x(2)^2 - 2^3 \\

&= x^3 - 6x^2 + 12x - 8.

\end{align}

File:binomial_theorem_visualisation.svg

For positive values of {{mvar|a}} and {{mvar|b}}, the binomial theorem with {{math|1=n = 2}} is the geometrically evident fact that a square of side {{math|a + b}} can be cut into a square of side {{mvar|a}}, a square of side {{mvar|b}}, and two rectangles with sides {{mvar|a}} and {{mvar|b}}. With {{math|1=n = 3}}, the theorem states that a cube of side {{math|a + b}} can be cut into a cube of side {{mvar|a}}, a cube of side {{mvar|b}}, three {{math|a × a × b}} rectangular boxes, and three {{math|a × b × b}} rectangular boxes.

In calculus, this picture also gives a geometric proof of the derivative $(x^n)\text{'}=nx^\{n-1\}:${{cite journal | last = Barth | first = Nils R.| title = Computing Cavalieri's Quadrature Formula by a Symmetry of the n-Cube | doi = 10.2307/4145193 | jstor = 4145193 | journal = The American Mathematical Monthly | volume = 111| issue = 9| pages = 811–813 | date=2004}} if one sets $a=x$ and $b=\backslash Delta\; x,$ interpreting {{mvar|b}} as an infinitesimal change in {{mvar|a}}, then this picture shows the infinitesimal change in the volume of an {{mvar|n}}-dimensional hypercube, $(x+\backslash Delta\; x)^n,$ where the coefficient of the linear term (in $\backslash Delta\; x$) is $nx^\{n-1\},$ the area of the {{mvar|n}} faces, each of dimension {{math|n − 1}}:

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

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

:"the infinitesimal rate of change in volume of an {{mvar|n}}-cube as side length varies is the area of {{mvar|n}} of its {{math|(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 $\backslash textstyle\{\backslash int\; x^\{n-1\}\backslash ,dx\; =\; \backslash tfrac\{1\}\{n\}\; x^n\}$ – see proof of Cavalieri's quadrature formula for details.

{{clear}}

{{Main|Binomial coefficient}}

The coefficients that appear in the binomial expansion are called binomial coefficients. These are usually written $\backslash tbinom\{n\}\{k\},$ and pronounced "{{mvar|n}} choose {{mvar|k}}".

The coefficient of {{math|x^n−ky^k}} is given by the formula

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

which is defined in terms of the factorial function {{math|n!}}. Equivalently, this formula can be written

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

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

The binomial coefficient $\backslash tbinom\; nk$ can be interpreted as the number of ways to choose {{mvar|k}} elements from an {{mvar|n}}-element set (a combination). This is related to binomials for the following reason: if we write {{math|1=(x + y)ⁿ}} as a product

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

then, according to the distributive law, there will be one term in the expansion for each choice of either {{mvar|x}} or {{mvar|y}} from each of the binomials of the product. For example, there will only be one term {{math|xⁿ}}, corresponding to choosing {{mvar|x}} from each binomial. However, there will be several terms of the form {{math|xⁿ⁻²y²}}, one for each way of choosing exactly two binomials to contribute a {{mvar|y}}. Therefore, after combining like terms, the coefficient of {{math|xⁿ⁻²y²}} will be equal to the number of ways to choose exactly {{math|2}} elements from an {{mvar|n}}-element set.

Expanding {{math|1=(x + y)ⁿ}} yields the sum of the {{math|2ⁿ}} products of the form {{math|1=e₁e₂ ... e_n}} where each {{math|e_i}} is {{mvar|x}} or {{mvar|y}}. Rearranging factors shows that each product equals {{math|x^n−ky^k}} for some {{mvar|k}} between {{math|0}} and {{mvar|n}}. For a given {{mvar|k}}, the following are proved equal in succession:

• the number of terms equal to {{math|1=x^n−ky^k}} in the expansion
• the number of {{mvar|n}}-character {{math|x,y}} strings having {{mvar|y}} in exactly {{mvar|k}} positions
• the number of {{mvar|k}}-element subsets of {{math|1={{mset|1, 2, ..., n}}}}
• $\backslash tbinom\{n\}\{k\},$ either by definition, or by a short combinatorial argument if one is defining $\backslash tbinom\{n\}\{k\}$ as $\backslash tfrac\{n!\}\{k!\; (n-k)!\}.$

This proves the binomial theorem.

The coefficient of {{math|xy²}} in

$$\backslash begin\{align\}$$

(x+y)^3 &= (x+y)(x+y)(x+y) \\

&= xxx + xxy + xyx + \underline{xyy} + yxx + \underline{yxy} + \underline{yyx} + yyy \\

&= x^3 + 3x^2y + \underline{3xy^2} + y^3

\end{align}

equals $\backslash tbinom\{3\}\{2\}=3$ because there are three {{math|x,y}} strings of length 3 with exactly two {{mvar|y}}'s, namely,

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

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

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

where each subset specifies the positions of the {{mvar|y}} in a corresponding string.

Induction yields another proof of the binomial theorem. When {{math|1=n = 0}}, both sides equal {{math|1}}, since {{math|1=x⁰ = 1}} and $\backslash tbinom\{0\}\{0\}=1.$ Now suppose that the equality holds for a given {{mvar|n}}; we will prove it for {{math|1=n + 1}}. For {{math|1=j, k ≥ 0}}, let {{math|1=[f(x, y)]_j,k}} denote the coefficient of {{math|1=x^jy^k}} in the polynomial {{math|1=f(x, y)}}. By the inductive hypothesis, {{math|1=(x + y)ⁿ}} is a polynomial in {{mvar|x}} and {{mvar|y}} such that {{math|1=[(x + y)ⁿ]_j,k}} is $\backslash tbinom\{n\}\{k\}$ if {{math|1=j + k = n}}, and {{mvar|0}} otherwise. The identity

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

shows that {{math|1=(x + y)ⁿ⁺¹}} is also a polynomial in {{mvar|x}} and {{mvar|y}}, and

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

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

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

by Pascal's identity.[http://proofs.wiki/Binomial_theorem Binomial theorem] – inductive proofs {{webarchive |url=https://web.archive.org/web/20150224130932/http://proofs.wiki/Binomial_theorem |date=February 24, 2015 }} On the other hand, if {{math|1=j + k ≠ n + 1}}, then {{math|1=(j – 1) + k ≠ n}} and {{math|1=j + (k – 1) ≠ n}}, so we get {{math|1=0 + 0 = 0}}. Thus

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

which is the inductive hypothesis with {{math|1=n + 1}} substituted for {{mvar|n}} and so completes the inductive step.

{{Main|Binomial series}}

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 {{mvar|r}}, one can define

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

where $(\backslash cdot)\_k$ is the Pochhammer symbol, here standing for a falling factorial. This agrees with the usual definitions when {{mvar|r}} is a nonnegative integer. Then, if {{mvar|x}} and {{mvar|y}} are real numbers with {{math|{{abs|x}} > {{abs|y}}}},This is to guarantee convergence. Depending on {{mvar|r}}, the series may also converge sometimes when {{math|1={{abs|x}} = {{abs|y}}}}. and {{mvar|r}} is any complex number, one has

$$\backslash begin\{align\}$$

(x+y)^r & =\sum_{k=0}^\infty {r \choose 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{align}

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

For example, {{math|1=r = 1/2}} gives the following series for the square root:

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

Taking {{math|1=r = −1}}, the generalized binomial series gives the geometric series formula, valid for {{math|{{abs|x}} < 1}}:

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

More generally, with {{math|1=r = −s}}, we have for {{math|{{abs|x}} < 1}}:{{cite web| url=https://mathworld.wolfram.com/NegativeBinomialSeries.html|title=Negative Binomial Series|website=Wolfram MathWorld|last=Weisstein|first=Eric W.}}

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

So, for instance, when {{math|1=s = 1/2}},

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

Replacing {{mvar|x}} with {{mvar|-x}} yields:

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

So, for instance, when {{math|1=s = 1/2}}, we have for {{math|{{abs|x}} < 1}}:

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

The generalized binomial theorem can be extended to the case where {{mvar|x}} and {{mvar|y}} are complex numbers. For this version, one should again assume {{math|{{abs|x}} > {{abs|y}}}} and define the powers of {{math|1=x + y}} and {{mvar|x}} using a holomorphic branch of log defined on an open disk of radius {{math|{{abs|x}}}} centered at {{mvar|x}}. The generalized binomial theorem is valid also for elements {{mvar|x}} and {{mvar|y}} of a Banach algebra as long as {{math|1=xy = yx}}, and {{mvar|x}} is invertible, and {{math|{{norm|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 {{mvar|c}}, define $x^\{(0)\}\; =\; 1$ and

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

for $n\; >\; 0.$ Then{{cite journal| url=https://cms.math.ca/publications/crux/issue/?volume=5&issue=2| title=Problem 352|first1=Dan|last1=Sokolowsky|first2=Basil C.|last2=Rennie|journal=Crux Mathematicorum|volume=5|issue=2|date=1979 | pages=55–56}}

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

The case {{math|1=c = 0}} recovers the usual binomial theorem.

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

• $\backslash deg\; p\_n\; =\; n$ for all $n$,
• $p\_0(0)\; =\; 1$, and
• $p\_n(x+y)\; =\; \backslash sum\_\{k=0\}^n\; \backslash binom\{n\}\{k\}\; p\_k(x)\; p\_\{n-k\}(y)$ for all $x$, $y$, and $n$.

An operator $Q$ on the space of polynomials is said to be the basis operator of the sequence $\backslash \{p\_n\backslash \}\_\{n=0\}^\backslash infty$ if $Qp\_0\; =\; 0$ and $Q\; p\_n\; =\; n\; p\_\{n-1\}$ for all $n\; \backslash geqslant\; 1$. A sequence $\backslash \{p\_n\backslash \}\_\{n=0\}^\backslash infty$ is binomial if and only if its basis operator is a Delta operator.{{cite book |last=Aigner |first=Martin |author-link=Martin Aigner |title=Combinatorial Theory |url=https://archive.org/details/combinatorialthe0000aign |url-access=limited |date=1979 |publisher=Springer |isbn=0-387-90376-3 |page=105 }} 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 .

{{Main|Multinomial theorem}}

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

$$(x\_1\; +\; x\_2\; +\; \backslash cdots\; +\; x\_m)^n\; =\; \backslash sum\_\{k\_1+k\_2+\backslash cdots\; +k\_m\; =\; n\}\; \backslash binom\{n\}\{k\_1,\; k\_2,\; \backslash ldots,\; k\_m\}\; x\_1^\{k\_1\}\; x\_2^\{k\_2\}\; \backslash cdots\; x\_m^\{k\_m\},$$

where the summation is taken over all sequences of nonnegative integer indices {{math|k₁}} through {{math|k_m}} such that the sum of all {{math|k_i}} is {{mvar|n}}. (For each term in the expansion, the exponents must add up to {{mvar|n}}). The coefficients $\backslash tbinom\{n\}\{k\_1,\backslash cdots,k\_m\}$ are known as multinomial coefficients, and can be computed by the formula

$$\backslash binom\{n\}\{k\_1,\; k\_2,\; \backslash ldots,\; k\_m\}\; =\; \backslash frac\{n!\}\{k\_1!\; \backslash cdot\; k\_2!\; \backslash cdots\; k\_m!\}.$$

Combinatorially, the multinomial coefficient $\backslash tbinom\{n\}\{k\_1,\backslash cdots,k\_m\}$ counts the number of different ways to partition an {{mvar|n}}-element set into disjoint subsets of sizes {{math|1=k₁, ..., 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

$$(x\_1+y\_1)^\{n\_1\}\backslash dotsm(x\_d+y\_d)^\{n\_d\}\; =\; \backslash sum\_\{k\_1=0\}^\{n\_1\}\backslash dotsm\backslash sum\_\{k\_d=0\}^\{n\_d\}\; \backslash binom\{n\_1\}\{k\_1\}\; x\_1^\{k\_1\}y\_1^\{n\_1-k\_1\}\; \backslash dotsc\; \backslash binom\{n\_d\}\{k\_d\}\; x\_d^\{k\_d\}y\_d^\{n\_d-k\_d\}.$$

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

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

{{Main|General Leibniz rule}}

The general Leibniz rule gives the {{mvar|n}}th derivative of a product of two functions in a form similar to that of the binomial theorem:{{cite book |last=Olver |first=Peter J. |author-link=Peter J. Olver |year=2000 |title=Applications of Lie Groups to Differential Equations |publisher=Springer |pages=318–319 |isbn=9780387950006 |url=https://books.google.com/books?id=sI2bAxgLMXYC&pg=PA318 }}

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

Here, the superscript {{math|(n)}} indicates the {{mvar|n}}th derivative of a function, $f^\{(n)\}(x)\; =\; \backslash tfrac\{d^n\}\{dx^n\}f(x)$. If one sets {{math|1=f(x) = e{{sup|ax}}}} and {{math|1=g(x) = e{{sup|bx}}}}, cancelling the common factor of {{math|e{{sup|(a + b)x}}}} from each term gives the ordinary binomial theorem.{{cite book |last1=Spivey |first1=Michael Z. |title=The Art of Proving Binomial Identities |date=2019 |publisher=CRC Press |isbn=978-1351215800 |page=71}}

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 $n=2$.{{cite journal|title=The Story of the Binomial Theorem|first=J. L.|last=Coolidge|journal=The American Mathematical Monthly| volume=56| issue=3|date=1949|pages=147–157|doi=10.2307/2305028|jstor = 2305028}} 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 {{mvar|k}} objects out of {{mvar|n}} 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 {{tmath|1= n = 4}} (probably obtained by listing all possibilities and counting them){{cite journal |last=Biggs |first=Norman L. |author-link=Norman L. Biggs |title=The roots of combinatorics |journal=Historia Mathematica |volume=6 |date=1979 |issue=2 |pages=109–136 |doi=10.1016/0315-0860(79)90074-0 |doi-access=free}} and a suggestion that higher combinations could likewise be found.{{cite journal |last=Datta |first=Bibhutibhushan |author-link=Bibhutibhushan Datta |url=https://archive.org/details/in.ernet.dli.2015.165748/page/n139/ |title=The Jaina School of Mathematics |journal=Bulletin of the Calcutta Mathematical Society |volume=27 |year=1929 |at=5. 115–145 (esp. 133–134) }} Reprinted as "The Mathematical Achievements of the Jainas" in {{cite book|editor-last=Chattopadhyaya |editor-first=Debiprasad |title=Studies in the History of Science in India |volume=2 |place=New Delhi |publisher=Editorial Enterprises |year=1982 |pages=684–716}} The Chandaḥśāstra by the Indian lyricist Piṅgala (3rd or 2nd century BC) somewhat crypically 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.{{cite journal |last=Bag |first=Amulya Kumar |title=Binomial theorem in ancient India |journal=Indian Journal of History of Science |volume=1 |number=1 |year=1966 |pages=68–74 |url=http://repository.ias.ac.in/70374/1/10-pub.pdf }} {{pb}} {{cite journal |last=Shah |first=Jayant |year=2013 |journal=Gaṇita Bhāratī |volume=35 |number=1–4 |pages=43–96 |title=A History of Piṅgala's Combinatorics |id={{ResearchGatePub|353496244}} }} ([https://ia800306.us.archive.org/19/items/Pingala/Pingala.pdf Preprint]) {{pb}} Survey sources: {{pb}} {{cite book |last=Edwards |first=A. W. F. |author-link=A. W. F. Edwards |year=1987 |chapter=The combinatorial numbers in India |title=Pascal's Arithmetical Triangle |place=London |publisher=Charles Griffin |isbn=0-19-520546-4 |chapter-url=https://archive.org/details/pascalsarithmeti0000edwa/page/27 |pages=27–33 |chapter-url-access=limited }} {{pb}} {{cite book |last=Divakaran |first=P. P. |year=2018 |title=The Mathematics of India: Concepts, Methods, Connections |chapter=Combinatorics |at=§5.5 {{pgs|135–140}} |publisher=Springer; Hindustan Book Agency |doi=10.1007/978-981-13-1774-3_5 |isbn=978-981-13-1773-6 }} {{pb}} {{cite book |last=Roy |first=Ranjan |author-link=Ranjan Roy |year=2021 |title=Series and Products in the Development of Mathematics |edition=2 |volume=1 |publisher=Cambridge University Press |chapter=The Binomial Theorem |at=Ch. 4, {{pgs|77–104}} |isbn=978-1-108-70945-3 |doi=10.1017/9781108709453.005 }} Varāhamihira (6th century AD) describes another method for computing combination counts by adding numbers in columns.{{cite journal |last=Gupta |first=Radha Charan |author-link=Radha Charan Gupta |title=Varāhamihira's Calculation of {{tmath|{}^nC_r}} and the Discovery of Pascal's Triangle |journal=Gaṇita Bhāratī |volume=14 |number=1–4 |year=1992 |pages=45–49 }} Reprinted in {{cite book |editor-last=Ramasubramanian |editor-first=K. |year=2019 |title=Gaṇitānanda |publisher=Springer |doi=10.1007/978-981-13-1229-8_29 |pages=285–289 }} By the 9th century at latest Indian mathematicians learned to express this as a product of fractions {{tmath| \tfrac{n}1 \times \tfrac{n - 1}2 \times \cdots \times \tfrac{n - k + 1}{n-k} }}, 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).{{r|gupta}}{{r|biggs}}{{cite book |year=1959|title=The Patiganita of Sridharacarya |editor-last=Shukla |editor-first=Kripa Shankar |editor-link= Kripa Shankar Shukla |publisher=Lucknow University |chapter-url=https://archive.org/details/Patiganita/page/n294/mode/1up |chapter=Combinations of Savours |at=Vyavahāras 1.9, {{pgs|97}} (text), {{pgs|58–59}} (translation) }}

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.{{cite journal |last=Yadegari |first=Mohammad |year=1980 |title=The Binomial Theorem: A Widespread Concept in Medieval Islamic Mathematics |journal=Historia Mathematica |volume=7 |issue=4 |pages=401–406 |doi=10.1016/0315-0860(80)90004-X |doi-access=free }}{{cite journal |last=Rashed |first=Roshdi |author-link=Roshdi Rashed |year=1972 |title=L'induction mathématique: al-Karajī, al-Samawʾal |journal=Archive for History of Exact Sciences |volume=9 |issue=1 |pages=1–21 |jstor=41133347 |doi=10.1007/BF00348537 |language=fr }} Translated into English by A. F. W. Armstrong in {{Cite book |last=Rashed |first=Roshdi |year=1994 |title=The Development of Arabic Mathematics: Between Arithmetic and Algebra |chapter=Mathematical Induction: al-Karajī and al-Samawʾal |chapter-url=https://archive.org/details/RoshdiRashedauth.TheDevelopmentOfArabicMathematicsBetweenArithmeticAndAlgebraSpringerNetherlands1994/page/n71/ |at=§1.4, {{pgs|62–81}} |doi=10.1007/978-94-017-3274-1_2 |publisher=Kluwer |isbn=0-7923-2565-6 |quote="The first formulation of the binomial and the table of binomial coefficients, to our knowledge, is to be found in a text by al-Karajī, cited by al-Samawʾal in al-Bāhir." }}{{Cite encyclopedia |title=Al-Karajī |encyclopedia=Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures |last=Sesiano |first=Jacques |editor-last=Selin |editor-first=Helaine |editor-link=Helaine Selin |year=1997 |publisher=Springer |doi=10.1007/978-94-017-1416-7_11 |isbn=978-94-017-1418-1 |pages=475–476 |quote=Another [lost work of Karajī's] contained the first known explanation of the arithmetical (Pascal's) triangle; the passage in question survived through al-Samawʾal's Bāhir (twelfth century) which heavily drew from the Badīʿ. }}

{{cite journal |last=Berggren |first=John Lennart |year=1985 |title=History of mathematics in the Islamic world: The present state of the art |journal=Review of Middle East Studies |volume=19 |number=1 |pages=9–33 |doi=10.1017/S0026318400014796 }} Republished in {{Cite book |title=From Alexandria, Through Baghdad |editor1-last=Sidoli |editor1-first=Nathan |editor2-last=Brummelen |editor2-first=Glen Van |editor2-link=Glen Van Brummelen |year=2014 |publisher=Springer |isbn=978-3-642-36735-9 |doi=10.1007/978-3-642-36736-6_4 |pages=51–71 |quote=[...] since the table of binomial coefficients had been previously found in such late works as those of al-Kāshī (fifteenth century) and Naṣīr al-Dīn al-Ṭūsī (thirteenth century), some had suggested that the table was a Chinese import. However, the use of the binomial coefficients by Islamic mathematicians of the eleventh century, in a context which had deep roots in Islamic mathematics, suggests strongly that the table was a local discovery – most probably of al-Karajī.}}

An explicit statement of the binomial theorem appears in al-Samawʾal's al-Bāhir (12th century), there credited to al-Karajī.{{r|yadegari}}{{r|rashed}} 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 {{tmath|1= n = 12}} and a rule for generating them equivalent to the recurrence relation {{tmath|1=\textstyle \binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k} }}.{{r|rashed}}{{MacTutor|id=Al-Karaji|title=Abu Bekr ibn Muhammad ibn al-Husayn Al-Karaji}} 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{{cite web | last = Landau | first = James A. | title = Historia Matematica Mailing List Archive: Re: [HM] Pascal's Triangle | work = Archives of Historia Matematica | format = mailing list email | access-date = 2007-04-13 | date = 1999-05-08 | url = http://archives.math.utk.edu/hypermail/historia/may99/0073.html | archive-date = 2021-02-24 | archive-url = https://web.archive.org/web/20210224081637/http://archives.math.utk.edu/hypermail/historia/may99/0073.html | url-status = dead }} 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.{{cite book |title=A History of Chinese Mathematics |chapter=Jia Xian and Liu Yi |last=Martzloff |first=Jean-Claude |translator-last=Wilson |translator-first=Stephen S. |publisher=Springer |year=1997 |orig-year=French ed. 1987 |isbn=3-540-54749-5 |page=142 |chapter-url=https://archive.org/details/historyofchinese0000mart_g2q8/page/142/mode/2up?&q=%22depends+on+the+binomial+expansion%22 |chapter-url-access=limited }}

In Europe, descriptions of the construction of Pascal's triangle can be found as early as Jordanus de Nemore's De arithmetica (13th century).{{cite journal |last=Hughes |first=Barnabas|year=1989 |title=The arithmetical triangle of Jordanus de Nemore |journal=Historia Mathematica |volume=16 |number=3 |pages=213–223 |doi=10.1016/0315-0860(89)90018-9 }} 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".{{cite book|title=History of mathematical thought|first=Morris| last=Kline| author-link=Morris Kline|page=273|publisher=Oxford University Press|year=1972}} 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.{{Cite book |last=Katz |first=Victor |author-link=Victor Katz |title=A History of Mathematics: An Introduction |edition=3rd |publisher=Addison-Wesley |year=2009 |orig-year=1993 |isbn=978-0-321-38700-4 |at=§ 14.3, {{pgs|487–497}} |chapter=Elementary Probability }}

By the early 17th century, some specific cases of the generalized binomial theorem, such as for $n=\backslash tfrac\{1\}\{2\}$, can be found in the work of Henry Briggs' Arithmetica Logarithmica (1624).{{r|stillwell}} Isaac Newton is generally credited with discovering the generalized binomial theorem, valid for any real exponent, in 1665, inspired by the work of John Wallis's Arithmetic Infinitorum and his method of interpolation.{{cite book |title=Elements of the History of Mathematics |date=1994 |first=N. |last=Bourbaki |author-link=Nicolas Bourbaki |translator=J. Meldrum |translator-link=John D. P. Meldrum |publisher=Springer |isbn=3-540-19376-6 |url-access=registration |url=https://archive.org/details/elementsofhistor0000bour}}{{Cite journal |last=Whiteside |first=D. T. |author-link=Tom Whiteside |date=1961 |title=Newton's Discovery of the General Binomial Theorem |url=https://www.cambridge.org/core/journals/mathematical-gazette/article/abs/newtons-discovery-of-the-general-binomial-theorem/19B5921B0248598CFB6441FCE085D113 |journal=The Mathematical Gazette |language=en |volume=45 |issue=353 |pages=175–180 |doi=10.2307/3612767 |jstor=3612767 }}{{r|stillwell}} A logarithmic version of the theorem for fractional exponents was discovered independently by James Gregory who wrote down his formula in 1670.{{cite book |last=Stillwell |first=John |author-link=John Stillwell |title=Mathematics and its history |date=2010 |publisher=Springer |isbn=978-1-4419-6052-8 |page=186 |edition=3rd}}

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,

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

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 {{math|cos(nx)}} and {{math|sin(nx)}}. For example, since

$$\backslash left(\backslash cos\; x\; +\; i\backslash sin\; x\backslash right)^2\; =\; \backslash cos^2\; x\; +\; 2i\; \backslash cos\; x\; \backslash sin\; x\; -\; \backslash sin^2\; x$$

= (\cos^2 x-\sin^2 x) + i(2\cos x\sin x),

But De Moivre's formula identifies the left side with $(\backslash cos\; x+i\backslash sin\; x)^2\; =\; \backslash cos(2x)+i\backslash sin(2x)$, so

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

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

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

De Moivre's formula yields

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

In general,

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

and

$$\backslash sin(nx)\; =\; \backslash sum\_\{k\backslash text\{\; odd\}\}\; (-1)^\{(k-1)/2\}\; \{n\; \backslash choose\; k\}\backslash cos^\{n-k\}\; x\; \backslash sin^k\; x.$$There are also similar formulas using Chebyshev polynomials.

The e (mathematical constant) is often defined by the formula

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

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

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

The {{mvar|k}}th term of this sum is

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

As {{math|n → ∞}}, the rational expression on the right approaches {{math|1}}, and therefore

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

This indicates that {{mvar|e}} can be written as a series:

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

Indeed, since each term of the binomial expansion is an increasing function of {{mvar|n}}, it follows from the monotone convergence theorem for series that the sum of this infinite series is equal to {{mvar|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 $\backslash \{X\_t\backslash \}\_\{t\backslash in\; S\}$ with probability of success $p\backslash in\; [0,1]$ all not happening is

:$P\backslash biggl(\backslash bigcap\_\{t\backslash in\; S\}\; X\_t^C\backslash biggr)\; =\; (1-p)^$

The binomial theorem is valid more generally for two elements {{math|x}} and {{math|y}} in a ring, or even a semiring, provided that {{math|1=xy = yx}}. For example, it holds for two {{math|n × n}} matrices, provided that those matrices commute; this is useful in computing powers of a matrix.{{cite book |last=Artin |first=Michael |author-link=Michael Artin |title=Algebra |edition=2nd |year=2011 |publisher=Pearson |at=equation (4.7.11)}}

The binomial theorem can be stated by saying that the polynomial sequence {{math|1={{mset|1, x, x², x³, ...}}}} is of binomial type.

{{portal|Mathematics}}

• 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

{{reflist|group=Note}}

{{reflist|30em}}

• {{cite book |last1=Graham |first1=Ronald |author1-link=Ronald Graham |last2=Knuth |first2=Donald |author2-link=Donald Knuth |first3=Oren |last3=Patashnik |author3-link=Oren Patashnik |title=Concrete Mathematics |publisher=Addison Wesley |year=1994 |edition=2nd |at=Ch. 5, {{pgs|153–256}} |chapter=Binomial Coefficients |isbn=978-0-201-55802-9 }}

{{Wikibooks|Combinatorics|Binomial Theorem|The Binomial Theorem}}

• {{SpringerEOM|id=Newton_binomial|first=E.D.|last= Solomentsev|title=Newton binomial}}
• [http://demonstrations.wolfram.com/BinomialTheorem/ Binomial Theorem] by Stephen Wolfram, and [http://demonstrations.wolfram.com/BinomialTheoremStepByStep/ "Binomial Theorem (Step-by-Step)"] by Bruce Colletti and Jeff Bryant, Wolfram Demonstrations Project, 2007.
• {{PlanetMath attribution

|urlname=InductiveProofOfBinomialTheorem |title=inductive proof of binomial theorem

}}

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Category:Factorial and binomial topics

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