Editing Binomial theorem

{{short description|Algebraic expansion of powers of a binomial}} 
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{{Image frame|width=215
|content=
<math>
\begin{array}{c}
 1 \\
 1 \quad 1 \\
 1 \quad 2 \quad 1 \\
 1 \quad 3 \quad 3 \quad 1 \\
 1 \quad 4 \quad 6 \quad 4 \quad 1 \\
 1 \quad 5 \quad 10 \quad 10 \quad 5 \quad 1 \\
 1 \quad 6 \quad 15 \quad 20 \quad 15 \quad 6 \quad 1 \\
 1 \quad 7 \quad 21 \quad 35 \quad 35 \quad 21 \quad 7 \quad 1
\end{array}
</math>
|caption=The [[binomial coefficient]] <math>\tbinom{n}{k}</math> appears as the {{mvar|k}}th entry in the {{mvar|n}}th row of [[Pascal's triangle]] (where the top is the 0th row <math>\tbinom{0}{0}</math>). Each entry is the sum of the two above it.}}
In [[elementary algebra]], the '''binomial theorem''' (or '''binomial expansion''') describes the [[Polynomial expansion|algebraic expansion]] of [[exponentiation|powers]] of a [[binomial (polynomial)|binomial]]. According to the theorem, the power {{tmath|\textstyle (x+y)^n}} expands into a [[polynomial]] with terms of the form {{tmath|\textstyle ax^ky^m }}, where the exponents {{tmath|k}} and {{tmath|m}} are [[nonnegative integer]]s satisfying {{tmath|1= k + m = n}} and the [[coefficient]] {{tmath|a}} of each term is a specific [[positive integer]] depending on {{tmath|n}} and {{tmath|k}}.  For example, for {{tmath|1= n = 4}},
<math display=block>(x+y)^4 = x^4 + 4 x^3y + 6 x^2 y^2 + 4 x y^3 + y^4. </math>

The coefficient {{tmath|a}} in each term {{tmath|\textstyle ax^ky^m }} is known as the [[binomial coefficient]] {{tmath|\tbinom nk}} or {{tmath|\tbinom{n}{m} }} (the two have the same value). These coefficients for varying {{tmath|n}} and {{tmath|k}} can be arranged to form [[Pascal's triangle]]. These numbers also occur in [[combinatorics]], where {{tmath|\tbinom nk}} gives the number of different [[combinations]] (i.e. subsets) of {{tmath|k}} [[element (mathematics)|elements]] that can be chosen from an {{tmath|n}}-element [[set (mathematics)|set]].  Therefore {{tmath|\tbinom nk}} is usually pronounced as "{{tmath|n}} choose {{tmath|k}}".

== Statement ==
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
<math display="block">(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}x^0 y^n,</math>
where each <math> \tbinom nk </math> is a positive integer known as a [[binomial coefficient]], defined as

<math display=block>\binom nk = \frac{n!}{k!\,(n-k)!} = \frac{n(n-1)(n-2)\cdots(n-k + 1)}{k(k-1)(k-2)\cdots2\cdot1}.</math>

This formula is also referred to as the '''binomial formula''' or the '''binomial identity'''. Using [[Capital-sigma notation|summation notation]], it can be written more concisely as
<math display="block">(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}.</math>

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, <math display=inline>\binom nk = \binom n{n-k}.</math>

A simple variant of the binomial formula is obtained by [[substitution (algebra)|substituting]] {{math|1}} for {{mvar|y}}, so that it involves only a single [[Variable (mathematics)|variable]]. In this form, the formula reads
<math display=block>\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}</math><!-- \vphantom{\Bigg)} works around a mediawiki scrollbar bug -->

== Examples ==

The first few cases of the binomial theorem are:
<math display="block">\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}</math>
In general, for the expansion of {{math|(''x'' + ''y'')<sup>''n''</sup>}} 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=''x''<sup>0</sup> = 1}});
* the exponents of {{mvar|y}} in the terms are {{math|0, 1, 2, ..., ''n'' − 1, ''n''}} (the first term implicitly contains {{math|1=''y''<sup>0</sup> = 1}});
* the coefficients form the {{mvar|n}}th row of Pascal's triangle;
* before combining like terms, there are {{math|2<sup>''n''</sup>}} terms {{math|''x''<sup>''i''</sup>''y''<sup>''j''</sup>}} in the expansion (not shown); 
* after combining like terms, there are {{math|''n'' + 1}} terms, and their coefficients sum to {{math|2<sup>''n''</sup>}}.
An example illustrating the last two points: <math display="block">\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}</math> with <math>1 + 3 + 3 + 1 = 2^3</math>.

A simple example with a specific positive value of {{math|''y''}}:
<math display="block">\begin{align}
(x+2)^3 &= x^3 + 3x^2(2) + 3x(2)^2 + 2^3 \\
&= x^3 + 6x^2 + 12x + 8.
\end{align}</math>

A simple example with a specific negative value of {{math|''y''}}:
<math display="block">\begin{align}
(x-2)^3 &= x^3 - 3x^2(2) + 3x(2)^2 - 2^3 \\
&= x^3 - 6x^2 + 12x - 8.
\end{align}</math>

=== Geometric explanation ===
[[File:binomial_theorem_visualisation.svg|thumb|300px|Visualisation of binomial expansion up to the 4th power]]
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]] <math>(x^n)'=nx^{n-1}:</math><ref name="barth2004">{{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}}</ref> if one sets <math>a=x</math> and <math>b=\Delta x,</math> 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]], <math>(x+\Delta x)^n,</math> where the coefficient of the linear term (in <math>\Delta x</math>) is <math>nx^{n-1},</math> the area of the {{mvar|n}} faces, each of dimension {{math|''n'' &minus; 1}}:
<math display="block">(x+\Delta x)^n = x^n + nx^{n-1}\Delta x + \binom{n}{2}x^{n-2}(\Delta x)^2 + \cdots.</math>
Substituting this into the [[definition of the derivative]] via a [[difference quotient]] and taking limits means that the higher order terms, <math>(\Delta x)^2</math> and higher, become negligible, and yields the formula <math>(x^n)'=nx^{n-1},</math> 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'' &minus; 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 <math>\textstyle{\int x^{n-1}\,dx = \tfrac{1}{n} x^n}</math> – see [[Cavalieri's quadrature formula#Proof|proof of Cavalieri's quadrature formula]] for details.<ref name="barth2004" />

{{clear}}

== Binomial coefficients ==
{{Main|Binomial coefficient}}
The coefficients that appear in the binomial expansion are called '''binomial coefficients'''. These are usually written <math>\tbinom{n}{k},</math> and pronounced "{{mvar|n}} choose {{mvar|k}}".

=== Formulas ===
The coefficient of {{math|''x''<sup>''n''−''k''</sup>''y''<sup>''k''</sup>}} is given by the formula
<math display="block">\binom{n}{k} = \frac{n!}{k! \; (n-k)!},</math>
which is defined in terms of the [[factorial]] function {{math|''n''!}}. Equivalently, this formula can be written
<math display="block">\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}</math>
with {{mvar|k}} factors in both the numerator and denominator of the [[Fraction (mathematics)|fraction]]. Although this formula involves a fraction, the binomial coefficient <math>\tbinom{n}{k}</math> is actually an [[integer]].

=== Combinatorial interpretation ===
The binomial coefficient <math> \tbinom nk </math> 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'')<sup>''n''</sup>}} as a [[Product (mathematics)|product]]
<math display="block">(x+y)(x+y)(x+y)\cdots(x+y),</math>
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|''x''<sup>''n''</sup>}}, corresponding to choosing {{mvar|x}} from each binomial. However, there will be several terms of the form {{math|''x''<sup>''n''−2</sup>''y''<sup>2</sup>}}, one for each way of choosing exactly two binomials to contribute a {{mvar|y}}. Therefore, after [[combining like terms]], the coefficient of {{math|''x''<sup>''n''−2</sup>''y''<sup>2</sup>}} will be equal to the number of ways to choose exactly {{math|2}} elements from an {{mvar|n}}-element set.

== Proofs ==

=== Combinatorial proof ===
Expanding {{math|1=(''x'' + ''y'')<sup>''n''</sup>}} yields the sum of the {{math|2<sup>''n''</sup>}} products of the form {{math|1=''e''<sub>1</sub>''e''<sub>2</sub> ... ''e''<sub>''n''</sub>}} where each {{math|''e''<sub>''i''</sub>}} is {{mvar|''x''}} or&nbsp;{{mvar|y}}. Rearranging factors shows that each product equals {{math|''x''<sup>''n''&minus;''k''</sup>''y''<sup>''k''</sup>}} for some {{mvar|k}} between {{math|0}} and&nbsp;{{mvar|n}}. For a given {{mvar|k}}, the following are proved equal in succession:
* the number of terms equal to {{math|1=''x''<sup>''n''−''k''</sup>''y''<sup>''k''</sup>}} 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''}}}}
* <math>\tbinom{n}{k},</math> either by definition, or by a short combinatorial argument if one is defining <math>\tbinom{n}{k}</math> as <math>\tfrac{n!}{k! (n-k)!}.</math>
This proves the binomial theorem.

==== Example ====
The coefficient of {{math|''xy''<sup>2</sup>}} in
<math display="block">\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}</math>
equals <math>\tbinom{3}{2}=3</math> because there are three {{math|''x'',''y''}} strings of length 3 with exactly two {{mvar|y}}'s, namely,
<math display="block">xyy, \; yxy, \; yyx,</math>
corresponding to the three 2-element subsets of {{math|{{mset|1, 2, 3}}}}, namely,
<math display="block">\{2,3\},\;\{1,3\},\;\{1,2\}, </math>
where each subset specifies the positions of the {{mvar|y}} in a corresponding string.

=== Inductive proof ===
[[mathematical induction|Induction]] yields another proof of the binomial theorem. When {{math|1=''n'' = 0}}, both sides equal {{math|1}}, since {{math|1=''x''<sup>0</sup> = 1}} and <math>\tbinom{0}{0}=1.</math>  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'')]<sub>''j'',''k''</sub>}} denote the coefficient of {{math|1=''x''<sup>''j''</sup>''y''<sup>''k''</sup>}} in the polynomial {{math|1=''f''(''x'', ''y'')}}.  By the inductive hypothesis, {{math|1=(''x'' + ''y'')<sup>''n''</sup>}} is a polynomial in {{mvar|x}} and {{mvar|y}} such that {{math|1=[(''x'' + ''y'')<sup>''n''</sup>]<sub>''j'',''k''</sub>}} is <math>\tbinom{n}{k}</math> if {{math|1=''j'' + ''k'' = ''n''}}, and {{mvar|0}} otherwise.  The identity
<math display="block"> (x+y)^{n+1} = x(x+y)^n + y(x+y)^n</math>
shows that {{math|1=(''x'' + ''y'')<sup>''n''+1</sup>}} is also a polynomial in {{mvar|x}} and {{mvar|y}}, and
<math display="block"> [(x+y)^{n+1}]_{j,k} = [(x+y)^n]_{j-1,k} + [(x+y)^n]_{j,k-1},</math>
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
<math display="block"> \binom{n}{k} + \binom{n}{k-1} = \binom{n+1}{k},</math>
by [[Pascal's identity]].<ref>[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 }}</ref> 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
<math display="block">(x+y)^{n+1} = \sum_{k=0}^{n+1} \binom{n+1}{k} x^{n+1-k} y^k,</math>
which is the inductive hypothesis with {{math|1=''n'' + 1}} substituted for {{mvar|n}} and so completes the inductive step.

== Generalizations ==

=== Newton's generalized binomial theorem ===
{{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 number|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
<math display="block">{r \choose k}=\frac{r(r-1) \cdots (r-k+1)}{k!} =\frac{(r)_k}{k!},</math><!--This is not the same as \frac{r!}{k!(r−k)!}. Please do not change it.-->
where <math>(\cdot)_k</math> 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''}}}},<ref name=convergence group=Note>This is to guarantee convergence. Depending on {{mvar|r}}, the series may also converge sometimes when {{math|1={{abs|''x''}} = {{abs|''y''}}}}.</ref> and {{mvar|r}} is any complex number, one has
<math display="block">\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}</math>

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:
<math display="block">\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.</math>

Taking {{math|1=''r'' = &minus;1}}, the generalized binomial series gives the [[Geometric series#Sum|geometric series formula]], valid for {{math|{{abs|''x''}} < 1}}:
<math display="block">(1+x)^{-1} = \frac{1}{1+x} = 1 - x + x^2 - x^3 + x^4 - x^5 + \cdots.</math>

More generally, with {{math|1=''r'' = −''s''}}, we have for {{math|{{abs|''x''}} < 1}}:<ref name=wolfram2>{{cite web| url=https://mathworld.wolfram.com/NegativeBinomialSeries.html|title=Negative Binomial Series|website=Wolfram MathWorld|last=Weisstein|first=Eric W.}}</ref>
<math display="block">\frac{1}{(1+x)^s} = \sum_{k=0}^\infty {-s \choose k} x^k = \sum_{k=0}^\infty {s+k-1 \choose k} (-1)^k x^k.</math>

So, for instance, when {{math|1=''s'' = 1/2}},
<math display="block">\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.</math>

Replacing {{mvar|x}} with {{mvar|-x}} yields:
<math display="block">\frac{1}{(1-x)^s} = \sum_{k=0}^\infty {s+k-1 \choose k} (-1)^k (-x)^k = \sum_{k=0}^\infty {s+k-1 \choose k} x^k.</math>

So, for instance, when {{math|1=''s'' = 1/2}}, we have for {{math|{{abs|''x''}} < 1}}:
<math display="block">\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.</math>

=== Further generalizations ===
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''}}}}<ref name=convergence group=Note /> and define the powers of {{math|1=''x'' + ''y''}} and {{mvar|x}} using a [[Holomorphic function|holomorphic]] [[complex logarithm|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 <math> x^{(0)} = 1 </math> and
<math display="block"> x^{(n)} = \prod_{k=1}^{n}[x+(k-1)c]</math>
for <math> n > 0.</math>  Then<ref name="Sokolowsky">{{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}}</ref>
<math display="block"> (a + b)^{(n)} = \sum_{k=0}^{n}\binom{n}{k}a^{(n-k)}b^{(k)}.</math>
The case {{math|1=''c'' = 0}} recovers the usual binomial theorem.

More generally, a sequence <math>\{p_n\}_{n=0}^\infty</math> of polynomials is said to be '''of binomial type''' if
* <math> \deg p_n = n </math> for all <math>n</math>,
* <math> p_0(0) = 1 </math>, and
* <math> p_n(x+y) = \sum_{k=0}^n \binom{n}{k} p_k(x) p_{n-k}(y) </math> for all <math>x</math>, <math>y</math>, and <math>n</math>.
An operator <math>Q</math> on the space of polynomials is said to be the ''basis operator'' of the sequence <math>\{p_n\}_{n=0}^\infty</math> if <math>Qp_0 = 0</math> and <math> Q p_n = n p_{n-1} </math> for all <math> n \geqslant 1 </math>. A sequence <math>\{p_n\}_{n=0}^\infty</math> is binomial if and only if its basis operator is a [[Delta operator]].<ref>{{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 }}</ref> Writing <math> E^a </math> for the shift by <math> a </math> operator, the Delta operators corresponding to the above "Pochhammer" families of polynomials are the backward difference <math> I - E^{-c} </math> for <math> c>0 </math>, the ordinary derivative for <math> c=0 </math>, and the forward difference <math> E^{-c} - I </math> for <math> c<0 </math>.

=== Multinomial theorem ===
{{Main|Multinomial theorem}}
The binomial theorem can be generalized to include powers of sums with more than two terms. The general version is

<math display="block">(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}, </math>

where the summation is taken over all sequences of nonnegative integer indices {{math|''k''<sub>1</sub>}} through {{math|''k''<sub>''m''</sub>}} such that the sum of all {{math|''k''<sub>''i''</sub>}} is&nbsp;{{mvar|n}}. (For each term in the expansion, the exponents must add up to&nbsp;{{mvar|n}}). The coefficients <math> \tbinom{n}{k_1,\cdots,k_m} </math> are known as multinomial coefficients, and can be computed by the formula
<math display="block"> \binom{n}{k_1, k_2, \ldots, k_m} = \frac{n!}{k_1! \cdot k_2! \cdots k_m!}.</math>

Combinatorially, the multinomial coefficient <math>\tbinom{n}{k_1,\cdots,k_m}</math> counts the number of different ways to [[Partition of a set|partition]] an {{mvar|n}}-element set into [[Disjoint sets|disjoint]] [[subset]]s of sizes {{math|1=''k''<sub>1</sub>, ..., ''k''<sub>''m''</sub>}}.

=== {{anchor|multi-binomial}} 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
<math display="block"> (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}. </math>

This may be written more concisely, by [[multi-index notation]], as
<math display="block"> (x+y)^\alpha = \sum_{\nu \le \alpha} \binom{\alpha}{\nu} x^\nu y^{\alpha - \nu}.</math>

=== General Leibniz rule ===
{{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:<ref>{{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 }}</ref> 
<math display="block">(fg)^{(n)}(x) = \sum_{k=0}^n \binom{n}{k} f^{(n-k)}(x) g^{(k)}(x).</math>

Here, the superscript {{math|(''n'')}} indicates the {{mvar|n}}th derivative of a function, <math>f^{(n)}(x) = \tfrac{d^n}{dx^n}f(x)</math>.  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.<ref>{{cite book |last1=Spivey |first1=Michael Z. |title=The Art of Proving Binomial Identities |date=2019 |publisher=CRC Press |isbn=978-1351215800 |page=71}}</ref>

==History==
Special cases of the binomial theorem were known since at least the 4th century BC when [[Greek mathematics|Greek mathematician]] [[Euclid]] mentioned the special case of the binomial theorem for exponent <math>n=2</math>.<ref name="Coolidge">{{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}}</ref> Greek mathematician [[Diophantus]] cubed various binomials, including <math>x-1</math>.<ref name="Coolidge" />  Indian mathematician [[Aryabhata]]'s method for finding cube roots, from around 510 AD, suggests that he knew the binomial formula for exponent <math>n=3</math>.<ref name="Coolidge" />

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 [[Jainism|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)<ref name=biggs>{{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}}</ref> and a suggestion that higher combinations could likewise be found.<ref>{{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}}</ref> 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 [[metre (poetry)|metre]]s 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]].<ref>{{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 }}</ref> [[Varāhamihira]] (6th century AD) describes another method for computing combination counts by adding numbers in columns.<ref name=gupta>{{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 }}</ref> 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 (mathematician)|Mahāvīra]]'s ''[[Gaṇita-sāra-saṅgraha]]'' (c. 850), and [[Bhāskara II]]'s ''Līlāvatī'' (12th century).{{r|gupta}}{{r|biggs}}<ref>{{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) }}</ref>

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.<ref name=yadegari>{{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 }}</ref><ref name=rashed>{{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''." }}</ref><ref>{{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''. }}</ref><ref>
{{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.}}</ref>
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}}<ref name=Karaji>{{MacTutor|id=Al-Karaji|title=Abu Bekr ibn Muhammad ibn al-Husayn Al-Karaji}}</ref> The Persian poet and mathematician [[Omar Khayyam]] was probably familiar with the formula to higher orders, although many of his mathematical works are lost.<ref name="Coolidge" /> The binomial expansions of small degrees were known in the 13th century mathematical works of [[Yang Hui]]<ref>{{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 }}</ref> and also [[Chu Shih-Chieh]].<ref name="Coolidge" />  Yang Hui attributes the method to a much earlier 11th century text of [[Jia Xian]], although those writings are now also lost.<ref>{{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 }}</ref>

In Europe, descriptions of the construction of Pascal's triangle can be found as early as [[Jordanus de Nemore]]'s ''De arithmetica'' (13th century).<ref>{{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 }}</ref> In 1544, [[Michael Stifel]] introduced the term "binomial coefficient" and showed how to use them to express <math>(1+x)^n</math> in terms of <math>(1+x)^{n-1}</math>, via "Pascal's triangle".<ref name=Kline>{{cite book|title=History of mathematical thought|first=Morris| last=Kline| author-link=Morris Kline|page=273|publisher=Oxford University Press|year=1972}}</ref> Other 16th century mathematicians including [[Niccolò Fontana Tartaglia]] and [[Simon Stevin]] also knew of it.<ref name=Kline /> 17th-century mathematician [[Blaise Pascal]] studied the eponymous triangle comprehensively in his ''Traité du triangle arithmétique''.<ref>{{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 }}</ref>

By the early 17th century, some specific cases of the generalized binomial theorem, such as for <math>n=\tfrac{1}{2}</math>, can be found in the work of [[Henry Briggs (mathematician)|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.<ref name=Kline /><ref>{{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}}</ref><ref name="Coolidge" /><ref>{{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 }}</ref>{{r|stillwell}} A logarithmic version of the theorem for fractional exponents was discovered independently by [[James Gregory (mathematician)|James Gregory]] who wrote down his formula in 1670.<ref name=stillwell>{{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}}</ref>

== Applications ==

=== Multiple-angle identities ===
For the [[complex numbers]] the binomial theorem can be combined with [[de Moivre's formula]] to yield [[List of trigonometric identities#Multiple-angle formulae|multiple-angle formulas]] for the [[sine]] and [[cosine]]. According to De Moivre's formula,
<math display="block">\cos\left(nx\right)+i\sin\left(nx\right) = \left(\cos x+i\sin x\right)^n.</math>

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
<math display="block">\left(\cos x + i\sin x\right)^2 = \cos^2 x + 2i \cos x \sin x - \sin^2 x
= (\cos^2 x-\sin^2 x) + i(2\cos x\sin x),</math>
But De Moivre's formula identifies the left side with <math>(\cos x+i\sin x)^2 = \cos(2x)+i\sin(2x)</math>, so
<math display="block">\cos(2x) = \cos^2 x - \sin^2 x \quad\text{and}\quad\sin(2x) = 2 \cos x \sin x,</math>
which are the usual double-angle identities. Similarly, since
<math display="block">\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,</math>
De Moivre's formula yields
<math display="block">\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.</math>
In general,
<math display="block">\cos(nx) = \sum_{k\text{ even}} (-1)^{k/2} {n \choose k}\cos^{n-k} x \sin^k x</math>
and
<math display="block">\sin(nx) = \sum_{k\text{ odd}} (-1)^{(k-1)/2} {n \choose k}\cos^{n-k} x \sin^k x.</math>There are also similar formulas using [[Chebyshev polynomials]].

=== Series for ''e'' ===
The [[e (mathematical constant)|number {{mvar|e}}]] is often defined by the formula
<math display="block">e = \lim_{n\to\infty} \left(1 + \frac{1}{n}\right)^n.</math>

Applying the binomial theorem to this expression yields the usual [[infinite series]] for {{mvar|e}}. In particular:
<math display="block">\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}.</math>

The {{mvar|k}}th term of this sum is
<math display="block">{n \choose k}\frac{1}{n^k} = \frac{1}{k!}\cdot\frac{n(n-1)(n-2)\cdots (n-k+1)}{n^k}</math>

As {{math|''n'' → ∞}}, the rational expression on the right approaches {{math|1}}, and therefore
<math display="block">\lim_{n\to\infty} {n \choose k}\frac{1}{n^k} = \frac{1}{k!}.</math>

This indicates that {{mvar|e}} can be written as a series:
<math display="block">e=\sum_{k=0}^\infty\frac{1}{k!}=\frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \cdots.</math>

Indeed, since each term of the binomial expansion is an [[Monotonic function|increasing function]] of {{mvar|n}}, it follows from the [[monotone convergence theorem]] for series that the sum of this infinite series is equal to&nbsp;{{mvar|e}}.

=== 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 <math>\{X_t\}_{t\in S}</math> with probability of success <math>p\in [0,1]</math> all not happening is
:<math> P\biggl(\bigcap_{t\in S} X_t^C\biggr) = (1-p)^{|S|} = \sum_{n=0}^{|S|} {|S| \choose n} (-p)^n.</math>
An upper bound for this quantity is <math> e^{-p|S|}.</math><ref>{{Cite book |title=Elements of Information Theory |chapter=Data Compression |last1=Cover |first1=Thomas M. |author1-link=Thomas M. Cover |last2=Thomas |first2=Joy A. |author2-link=Joy A. Thomas |date=1991 |publisher=Wiley |isbn=9780471062592 |at=Ch. 5, {{pgs|78–124}} |doi=10.1002/0471200611.ch5}}<!-- a specific page number would be helpful. previously this citation noted p. 320 but that's not in this chapter. --> </ref>

== In abstract algebra ==

The binomial theorem is valid more generally for two elements {{math|''x''}} and {{math|''y''}} in a [[Ring_(mathematics)|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.<ref>{{cite book |last=Artin |first=Michael |author-link=Michael Artin |title=Algebra |edition=2nd |year=2011 |publisher=Pearson |at=equation (4.7.11)}}</ref>

The binomial theorem can be stated by saying that the [[polynomial sequence]]  {{math|1={{mset|1, ''x'', ''x''<sup>2</sup>, ''x''<sup>3</sup>, ...}}}} is of [[binomial type]].

== See also ==
{{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]]
* [[Gaussian binomial coefficient#q-binomial theorem|q-binomial theorem]]

== Notes ==
{{reflist|group=Note}}

== References ==
{{reflist|30em}}

== Further reading ==
* {{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 }}

== External links ==
{{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
}}
{{Calculus topics}}
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[[Category:Factorial and binomial topics]]
[[Category:Theorems about polynomials]]
[[Category:Articles containing proofs]]