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Binomial theorem

In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial $(x + y)^{n}$ into a sum involving terms of the form $ax^{b}y^{c}$, where the exponents $k$ and $n$ are nonnegative integers with $b + c = n$, and the coefficient $b$ of each term is a specific positive integer depending on $c$ and $a$. For example, for $n = 4$, $$(x+y)^4 = x^4 + 4 x^3y + 6 x^2 y^2 + 4 x y^3 + y^4. $$

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

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 CE, suggests that he knew the binomial formula for exponent $$n=3$$.

Binomial coefficients, as combinatorial quantities expressing the number of ways of selecting $n$ objects out of $b$ without replacement, were of interest to ancient Indian mathematicians. The earliest known reference to this combinatorial problem is the Chandaḥśāstra by the Indian lyricist Pingala (c. 200 BC), which contains a method for its solution. The commentator Halayudha from the 10th century AD explains this method. By the 6th century AD, the Indian mathematicians probably knew how to express this as a quotient $\frac{n!}{(n-k)!k!}$, and a clear statement of this rule can be found in the 12th century text Lilavati by Bhaskara.

The first known formulation of the binomial theorem and the table of binomial coefficients appears in a work by Al-Karaji, quoted by Al-Samaw'al in his "al-Bahir". Al-Karaji described the triangular pattern of the binomial coefficients and also provided a mathematical proof of both the binomial theorem and Pascal's triangle, using an early form of mathematical induction. The Persian poet and mathematician Omar Khayyam was probably familiar with the formula to higher orders, although many of his mathematical works are lost. The binomial expansions of small degrees were known in the 13th century mathematical works of Yang Hui and also Chu Shih-Chieh. Yang Hui attributes the method to a much earlier 11th century text of Jia Xian, although those writings are now also lost.

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". Blaise Pascal studied the eponymous triangle comprehensively in his Traité du triangle arithmétique. However, the pattern of numbers was already known to the European mathematicians of the late Renaissance, including Stifel, Niccolò Fontana Tartaglia, and Simon Stevin.

Isaac Newton is generally credited with discovering the generalized binomial theorem, valid for any real exponent, in 1665. It was discovered independently in 1670 by James Gregory.

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

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

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

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

A simple variant of the binomial formula is obtained by substituting $1$ for $x$, so that it involves only a single variable. In this form, the formula reads $$\begin{align} (x+1)^n &= {n \choose 0}x^0 + {n \choose 1}x^1 + {n \choose 2}x^2 + \cdots + {n \choose {n-1}}x^{n-1} + {n \choose n}x^n \\[4mu] &= \sum_{k=0}^n {n \choose k}x^k. \vphantom{\Bigg)} \end{align}$$

Examples
Here are the first few cases of the binomial theorem: $$\begin{align} (x+y)^0 & = 1, \\[8pt] (x+y)^1 & = x + y, \\[8pt] (x+y)^2 & = x^2 + 2xy + y^2, \\[8pt] (x+y)^3 & = x^3 + 3x^2y + 3xy^2 + y^3, \\[8pt] (x+y)^4 & = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4, \\[8pt] (x+y)^5 & = x^5 + 5x^4y + 10x^3y^2 + 10x^2y^3 + 5xy^4 + y^5, \\[8pt] (x+y)^6 & = x^6 + 6x^5y + 15x^4y^2 + 20x^3y^3 + 15x^2y^4 + 6xy^5 + y^6, \\[8pt] (x+y)^7 & = x^7 + 7x^6y + 21x^5y^2 + 35x^4y^3 + 35x^3y^4 + 21x^2y^5 + 7xy^6 + y^7, \\[8pt] (x+y)^8 & = x^8 + 8x^7y + 28x^6y^2 + 56x^5y^3 + 70x^4y^4 + 56x^3y^5 + 28x^2y^6 + 8xy^7 + y^8. \end{align}$$ In general, for the expansion of $(x + y)^{n}$ on the right side in the $y$th row (numbered so that the top row is the 0th row): An example illustrating the last two points: $$\begin{align} (x+y)^3 & = xxx + xxy + xyx + xyy + yxx + yxy + yyx + yyy & (2^3 \text{ terms}) \\ & = x^3 + 3x^2y + 3xy^2 + y^3 & (3 + 1 \text{ terms}) \end{align}$$ with $$1 + 3 + 3 + 1 = 2^3$$.
 * the exponents of $y$ in the terms are $n, n − 1, ..., 2, 1, 0$ (the last term implicitly contains $x^{0} = 1$);
 * the exponents of $n$ in the terms are $0, 1, 2, ..., n − 1, n$ (the first term implicitly contains $y^{0} = 1$);
 * the coefficients form the $x$th row of Pascal's triangle;
 * before combining like terms, there are $2^{n}$ terms $x^{i}y^{j}$ in the expansion (not shown);
 * after combining like terms, there are $n + 1$ terms, and their coefficients sum to $2^{n}$.

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

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

Geometric explanation
For positive values of $y$ and $n$, the binomial theorem with $n = 2$ is the geometrically evident fact that a square of side $a + b$ can be cut into a square of side $a$, a square of side $b$, and two rectangles with sides $a$ and $b$. With $n = 3$, the theorem states that a cube of side $a + b$ can be cut into a cube of side $a$, a cube of side $b$, three $a × a × b$ rectangular boxes, and three $a × b × b$ rectangular boxes.

In calculus, this picture also gives a geometric proof of the derivative $$(x^n)'=nx^{n-1}:$$ if one sets $$a=x$$ and $$b=\Delta x,$$ interpreting $a$ as an infinitesimal change in $b$, then this picture shows the infinitesimal change in the volume of an $b$-dimensional hypercube, $$(x+\Delta x)^n,$$ where the coefficient of the linear term (in $$\Delta x$$) is $$nx^{n-1},$$ the area of the $a$ faces, each of dimension $n − 1$: $$(x+\Delta x)^n = x^n + nx^{n-1}\Delta x + \binom{n}{2}x^{n-2}(\Delta x)^2 + \cdots.$$ Substituting this into the definition of the derivative via a difference quotient and taking limits means that the higher order terms, $$(\Delta x)^2$$ and higher, become negligible, and yields the formula $$(x^n)'=nx^{n-1},$$ interpreted as
 * "the infinitesimal rate of change in volume of an $n$-cube as side length varies is the area of $n$ 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 $$\textstyle{\int x^{n-1}\,dx = \tfrac{1}{n} x^n}$$ – 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 $$\tbinom{n}{k},$$ and pronounced "$n$ choose $n$".

Formulas
The coefficient of $x^{n−k}y^{k}$ is given by the formula $$\binom{n}{k} = \frac{n!}{k! \; (n-k)!},$$ which is defined in terms of the factorial function $n!$. Equivalently, this formula can be written $$\binom{n}{k} = \frac{n (n-1) \cdots (n-k+1)}{k (k-1) \cdots 1} = \prod_{\ell=1}^k \frac{n-\ell+1}{\ell} = \prod_{\ell=0}^{k-1} \frac{n-\ell}{k - \ell}$$ with $n$ factors in both the numerator and denominator of the fraction. Although this formula involves a fraction, the binomial coefficient $$\tbinom{n}{k}$$ is actually an integer.

Combinatorial interpretation
The binomial coefficient $$ \tbinom nk $$ can be interpreted as the number of ways to choose $k$ elements from an $k$-element set. 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 $k$ or $n$ from each of the binomials of the product. For example, there will only be one term $x^{n}$, corresponding to choosing $x$ from each binomial. However, there will be several terms of the form $x^{n−2}y^{2}$, one for each way of choosing exactly two binomials to contribute a $y$. Therefore, after combining like terms, the coefficient of $x^{n−2}y^{2}$ will be equal to the number of ways to choose exactly $2$ elements from an $x$-element set.

Combinatorial proof
Expanding $(x + y)^{n}$ yields the sum of the $2^{n}$ products of the form $e_{1}e_{2} ... e_{n}$ where each $e_{i}$ is $y$ or $n$. Rearranging factors shows that each product equals $x^{n−k}y^{k}$ for some $x$ between $0$ and $y$. For a given $k$, the following are proved equal in succession: This proves the binomial theorem.
 * the number of terms equal to $x^{n−k}y^{k}$ in the expansion
 * the number of $n$-character $x,y$ strings having $k$ in exactly $n$ positions
 * the number of $y$-element subsets of $\{1, 2, ..., n\}$
 * $$\tbinom{n}{k},$$ either by definition, or by a short combinatorial argument if one is defining $$\tbinom{n}{k}$$ as $$\tfrac{n!}{k! (n-k)!}.$$

Example
The coefficient of $xy^{2}$ in $$\begin{align} (x+y)^3 &= (x+y)(x+y)(x+y) \\ &= xxx + xxy + xyx + \underline{xyy} + yxx + \underline{yxy} + \underline{yyx} + yyy \\ &= x^3 + 3x^2y + \underline{3xy^2} + y^3 \end{align}$$ equals $$\tbinom{3}{2}=3$$ because there are three $x,y$ strings of length 3 with exactly two $k$'s, namely, $$xyy, \; yxy, \; yyx,$$ corresponding to the three 2-element subsets of $\{1, 2, 3\}$, namely, $$\{2,3\},\;\{1,3\},\;\{1,2\}, $$ where each subset specifies the positions of the $k$ in a corresponding string.

Inductive proof
Induction yields another proof of the binomial theorem. When $n = 0$, both sides equal $1$, since $x^{0} = 1$ and $$\tbinom{0}{0}=1.$$ Now suppose that the equality holds for a given $y$; we will prove it for $n + 1$. For $j, k ≥ 0$, let $[f(x, y)]_{j,k}$ denote the coefficient of $x^{j}y^{k}$ in the polynomial $f(x, y)$. By the inductive hypothesis, $(x + y)^{n}$ is a polynomial in $y$ and $n$ such that $[(x + y)^{n}]_{j,k}$ is $$\tbinom{n}{k}$$ if $j + k = n$, and $x$ 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 $y$ and $0$, 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 $$ \binom{n}{k} + \binom{n}{k-1} = \binom{n+1}{k},$$ 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 $$(x+y)^{n+1} = \sum_{k=0}^{n+1} \binom{n+1}{k} x^{n+1-k} y^k,$$ which is the inductive hypothesis with $n + 1$ substituted for $x$ and so completes the inductive step.

Newton's generalized binomial theorem
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 $y$, one can define $${r \choose k}=\frac{r(r-1) \cdots (r-k+1)}{k!} =\frac{(r)_k}{k!},$$ where $$(\cdot)_k$$ is the Pochhammer symbol, here standing for a falling factorial. This agrees with the usual definitions when $n$ is a nonnegative integer. Then, if $r$ and $r$ are real numbers with $|x| > |y|$, and $x$ is any complex number, one has $$\begin{align} (x+y)^r & =\sum_{k=0}^\infty {r \choose k} x^{r-k} y^k \\ &= x^r + r x^{r-1} y + \frac{r(r-1)}{2!} x^{r-2} y^2 + \frac{r(r-1)(r-2)}{3!} x^{r-3} y^3 + \cdots. \end{align}$$

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

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

Taking $r = 1/2$, the generalized binomial series gives the geometric series formula, valid for $r = −1$: $$(1+x)^{-1} = \frac{1}{1+x} = 1 - x + x^2 - x^3 + x^4 - x^5 + \cdots.$$

More generally, with $|x| < 1$, we have for $r = −s$: $$\frac{1}{(1+x)^s} = \sum_{k=0}^\infty {-s \choose k} x^k = \sum_{k=0}^\infty {s+k-1 \choose k} (-1)^k x^k.$$

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

Replacing $r$ with $r$ yields: $$\frac{1}{(1-x)^s} = \sum_{k=0}^\infty {s+k-1 \choose k} (-1)^k (-x)^k = \sum_{k=0}^\infty {s+k-1 \choose k} x^k.$$

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

Further generalizations
The generalized binomial theorem can be extended to the case where $r$ and $x$ are complex numbers. For this version, one should again assume $|x| < 1$ and define the powers of $|x| > |y|$ and $-x$ using a holomorphic branch of log defined on an open disk of radius $x + y$ centered at $x$. The generalized binomial theorem is valid also for elements $y$ and $x$ of a Banach algebra as long as $|x|$, and $x$ is invertible, and $xy = yx$.

A version of the binomial theorem is valid for the following Pochhammer symbol-like family of polynomials: for a given real constant $x$, define $$ x^{(0)} = 1 $$ and $$ x^{(n)} = \prod_{k=1}^{n}[x+(k-1)c]$$ for $$ n > 0.$$ Then $$ (a + b)^{(n)} = \sum_{k=0}^{n}\binom{n}{k}a^{(n-k)}b^{(k)}.$$ The case $\|y/x\| < 1$ recovers the usual binomial theorem.

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

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

$$(x_1 + x_2 + \cdots + x_m)^n = \sum_{k_1+k_2+\cdots +k_m = n} \binom{n}{k_1, k_2, \ldots, k_m} x_1^{k_1} x_2^{k_2} \cdots x_m^{k_m}, $$

where the summation is taken over all sequences of nonnegative integer indices $c = 0$ through $k_{1}$ such that the sum of all $k_{m}$ is $y$. (For each term in the expansion, the exponents must add up to $x$). The coefficients $$ \tbinom{n}{k_1,\cdots,k_m} $$ are known as multinomial coefficients, and can be computed by the formula $$ \binom{n}{k_1, k_2, \ldots, k_m} = \frac{n!}{k_1! \cdot k_2! \cdots k_m!}.$$

Combinatorially, the multinomial coefficient $$\tbinom{n}{k_1,\cdots,k_m}$$ counts the number of different ways to partition an $c$-element set into disjoint subsets of sizes $k_{i}$.

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 $$ (x_1+y_1)^{n_1}\dotsm(x_d+y_d)^{n_d} = \sum_{k_1=0}^{n_1}\dotsm\sum_{k_d=0}^{n_d} \binom{n_1}{k_1} x_1^{k_1}y_1^{n_1-k_1} \dotsc \binom{n_d}{k_d} x_d^{k_d}y_d^{n_d-k_d}. $$

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

General Leibniz rule
The general Leibniz rule gives the $n$th derivative of a product of two functions in a form similar to that of the binomial theorem: $$(fg)^{(n)}(x) = \sum_{k=0}^n \binom{n}{k} f^{(n-k)}(x) g^{(k)}(x).$$

Here, the superscript $k_{1}, ..., k_{m}$ indicates the $n$th derivative of a function, $$f^{(n)}(x) = \tfrac{d^n}{dx^n}f(x)$$. If one sets $(n)$ and $f(x) = eax$, cancelling the common factor of $g(x) = ebx$ from each term gives the ordinary binomial theorem.

Multiple-angle identities
For the complex numbers the binomial theorem can be combined with de Moivre's formula to yield multiple-angle formulas for the sine and cosine. According to De Moivre's formula, $$\cos\left(nx\right)+i\sin\left(nx\right) = \left(\cos x+i\sin x\right)^n.$$

Using the binomial theorem, the expression on the right can be expanded, and then the real and imaginary parts can be taken to yield formulas for $e(a + b)x$ and $cos(nx)$. For example, since $$\left(\cos x + i\sin x\right)^2 = \cos^2 x + 2i \cos x \sin x - \sin^2 x = (\cos^2 x-\sin^2 x) + i(2\cos x\sin x),$$ But De Moivre's formula identifies the left side with $$(\cos x+i\sin x)^2 = \cos(2x)+i\sin(2x)$$, so $$\cos(2x) = \cos^2 x - \sin^2 x \quad\text{and}\quad\sin(2x) = 2 \cos x \sin x,$$ which are the usual double-angle identities. Similarly, since $$\left(\cos x + i\sin x\right)^3 = \cos^3 x + 3i \cos^2 x \sin x - 3 \cos x \sin^2 x - i \sin^3 x,$$ De Moivre's formula yields $$\cos(3x) = \cos^3 x - 3 \cos x \sin^2 x \quad\text{and}\quad \sin(3x) = 3\cos^2 x \sin x - \sin^3 x.$$ In general, $$\cos(nx) = \sum_{k\text{ even}} (-1)^{k/2} {n \choose k}\cos^{n-k} x \sin^k x$$ and $$\sin(nx) = \sum_{k\text{ odd}} (-1)^{(k-1)/2} {n \choose k}\cos^{n-k} x \sin^k x.$$There are also similar formulas using Chebyshev polynomials.

Series for e
The number $n$ is often defined by the formula $$e = \lim_{n\to\infty} \left(1 + \frac{1}{n}\right)^n.$$

Applying the binomial theorem to this expression yields the usual infinite series for $n$. In particular: $$\left(1 + \frac{1}{n}\right)^n = 1 + {n \choose 1}\frac{1}{n} + {n \choose 2}\frac{1}{n^2} + {n \choose 3}\frac{1}{n^3} + \cdots + {n \choose n}\frac{1}{n^n}.$$

The $n$th term of this sum is $${n \choose k}\frac{1}{n^k} = \frac{1}{k!}\cdot\frac{n(n-1)(n-2)\cdots (n-k+1)}{n^k}$$

As $sin(nx)$, the rational expression on the right approaches $n → ∞$, and therefore $$\lim_{n\to\infty} {n \choose k}\frac{1}{n^k} = \frac{1}{k!}.$$

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

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

Probability
The binomial theorem is closely related to the probability mass function of the negative binomial distribution. The probability of a (countable) collection of independent Bernoulli trials $$\{X_t\}_{t\in S}$$ with probability of success $$p\in [0,1]$$ all not happening is
 * $$ P\left(\bigcap_{t\in S} X_t^C\right) = (1-p)^{|S|} = \sum_{n=0}^{|S|} {|S| \choose n} (-p)^n.$$

An upper bound for this quantity is $$ e^{-p|S|}.$$

In abstract algebra
The binomial theorem is valid more generally for two elements $1$ and $x$ in a ring, or even a semiring, provided that $y$. For example, it holds for two $xy = yx$ 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 $n × n$ is of binomial type.

In popular culture

 * The binomial theorem is mentioned in the Major-General's Song in the comic opera The Pirates of Penzance.
 * Professor Moriarty is described by Sherlock Holmes as having written a treatise on the binomial theorem.
 * The Portuguese poet Fernando Pessoa, using the heteronym Álvaro de Campos, wrote that "Newton's Binomial is as beautiful as the Venus de Milo. The truth is that few people notice it."
 * In the 2014 film The Imitation Game, Alan Turing makes reference to Isaac Newton's work on the binomial theorem during his first meeting with Commander Denniston at Bletchley Park.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Binomial theorem


text:
Balboa

Balboa may refer to:

Boats

 * Balboa 16, an American sailboat design
 * Balboa 20, an American sailboat design
 * Balboa 21, an American sailboat design
 * Balboa 22, an American sailboat design
 * Balboa 23, an American sailboat design
 * Balboa 24, an American sailboat design

Places

 * Balboa, Cauca, a town and municipality in Colombia
 * Balboa, León, a Spanish village and municipality
 * Balboa, Panama, a port city in Panama
 * Balboa District of Panamá Province in Panama
 * Balboa, Risaralda, a town and municipality in Colombia
 * Balboa (Los Angeles Metro station), on the Los Angeles Metro Orange Line
 * Balboa (lunar crater), located near the western limb of the Moon
 * Balboa High School (California), an American public high school of San Francisco, California
 * Balboa Island, Newport Beach, California, a harborside community in Newport Beach
 * Balboa Park (disambiguation), any of several
 * Balboa Peninsula, Newport Beach, California, a neighborhood of the city of Newport Beach
 * Naval Medical Center San Diego, a US Navy medical treatment facility informally known as "Balboa Hospital"

People

 * Vasco Núñez de Balboa (c. 1475 – 1519), a Spanish explorer, governor, and conquistador
 * Javier Balboa (born 1985), football midfielder
 * Marcelo Balboa (born 1967), American former soccer defender
 * Álex Balboa (born 2001), football midfielder

Entertainment

 * Balboa (dance), a close embrace style of swing dancing
 * Balboa (band), a hardcore band from Philadelphia
 * Balboa County, a fictional county setting for the Veronica Mars television series
 * Rocky Balboa, protagonist of the Rocky film series
 * Balboa, the merged tribe name in Survivor: Pearl Islands
 * Balboa, a probe featured in Alien Planet

Other uses

 * Balboa (bug), a genus of dirt-colored seed bugs
 * Balboa Line, a former train route in California, U.S.
 * Panamanian balboa, the official currency of Panama

title: Balboa


text:
Boxing Day

Boxing Day is a holiday celebrated after Christmas Day, occurring on the second day of Christmastide (26 December). Boxing Day was once a day to donate gifts to those in need, but it has evolved to become a part of Christmas festivities, with many people choosing to shop for deals on Boxing Day. It originated in the United Kingdom and is celebrated in several Commonwealth nations. The attached bank holiday or public holiday may take place on 28 December if necessary to ensure it falls on a weekday. Boxing Day is also concurrent with the Christian festival Saint Stephen's Day.

In parts of Europe, such as several regions of Spain, the Czech Republic, Germany, Austria, Hungary, Netherlands, Italy, Poland, Slovakia, Slovenia, Croatia, Denmark, Finland, Sweden, Belgium, Norway, and the Republic of Ireland, 26 December is Saint Stephen's Day, which is considered the second day of Christmas.

Etymology
There are competing theories for the origins of the term, none of which are definitive.

The European tradition of giving money and other gifts to those in need, or in service positions, has been dated to the Middle Ages, but the exact origin is unknown; it may reference the alms box placed in the narthex of Christian churches to collect donations for the poor. The tradition may come from a custom in the late Roman and early Christian era, wherein alms boxes placed in churches were used to collect special offerings tied to the Feast of Saint Stephen, which, in the Western Christian Churches, falls on the same day as Boxing Day, the second day of Christmastide. On this day, it is customary in some localities for the alms boxes to be opened and distributed to the poor.

The Oxford English Dictionary gives the earliest attestation from Britain in 1743, defining it as "the day after Christmas day", and saying "traditionally on this day tradespeople, employees, etc., would receive presents or gratuities (a "Christmas box") from their customers or employers."

The term "Christmas box" dates back to the 17th century, and among other things meant:

"A present or gratuity given at Christmas: In Great Britain, usually confined to gratuities given to those who are supposed to have a vague claim upon the donor for services rendered to him as one of the general public by whom they are employed and paid, or as a customer of their legal employer; the undefined theory being that as they have done offices for this person, for which he has not directly paid them, some direct acknowledgement is becoming at Christmas."

In Britain, it was a custom for tradesmen to collect "Christmas boxes" of money or presents on the first weekday after Christmas as thanks for good service throughout the year. This is mentioned in Samuel Pepys' diary entry for 19 December 1663. This custom is linked to an older British tradition in which the servants of the wealthy were allowed the next day to visit their families since they would have had to serve their masters on Christmas Day. The employers would give each servant a box to take home containing gifts, bonuses, and sometimes leftover food. Until the late 20th century, there continued to be a tradition among many in the UK to give a Christmas gift, usually cash, to vendors, although not on Boxing Day, as many would not work on that day.

Date
Saint Stephen's Day, a religious holiday, also falls on 26 December.

In the United Kingdom, Boxing Day could not fall on a Sunday 26 December. Instead, Boxing Day would be celebrated on Monday 27 December, with the preceding Sunday called Christmas Sunday. This rule was independent of the rule of bank holidays being taken in lieu. The rule of no Boxing Day on a Sunday appeared to quietly change around the time that the Sunday Trading Act 1994 was enacted, with the last Boxing Day to land on 27 December being in 1993.

Unlike the contemporary understanding of Boxing Day itself, the associated bank holiday or public holiday always falls on a weekday. When 25 December falls on a Saturday and 26 December falls on a Sunday, the Christmas Day substitute holiday is observed on Monday 27 December, with the Boxing Day substitute holiday observed on Tuesday 28 December. When Christmas Day is a Sunday, the Boxing Day holiday is still observed on Monday 26 December, with the substitute holiday for Christmas Day observed on Tuesday 27 December.

The Banking and Financial Dealings Act 1971, which regulates UK bank holidays, does not officially name the 26 December bank holiday as Boxing Day, but states that it falls on "26th December, if it be not a Sunday."

Status by country

 * In Australia, Boxing Day is a public holiday in all jurisdictions except the state of South Australia, where a public holiday known as Proclamation Day is celebrated on the first weekday after Christmas Day or the Christmas Day holiday. Both the Boxing Day Test cricket match held at the Melbourne Cricket Ground and The Sydney to Hobart Yacht Race begin on Boxing Day.
 * In Canada, Boxing Day (le Lendemain de Noël) is a federal statutory holiday. Government offices, banks, and postal services are closed. In Ontario and other Canadian provinces, it is a provincial statutory holiday.
 * In Hong Kong, despite the transfer of sovereignty from the UK to China in 1997, Boxing Day is a general holiday as the first weekday after Christmas.
 * In Ireland, when the entire island was part of the United Kingdom, the Bank Holidays Act 1871 established the feast day of Saint Stephen as a non-moveable public holiday on 26 December. Following partition in 1920, Northern Ireland reverted to the British name, Boxing Day. In County Donegal, particularly in East Donegal and Inishowen, the day is also popularly known as Boxing Day.
 * In New Zealand, Boxing Day is a statutory holiday. On these holidays, people who must work receive $1 1/2$ times their salaries, and a day in lieu is provided to employees who work.
 * In Nigeria, Boxing Day is a public holiday for working people and students. When it falls on a Saturday or Sunday, there is always a holiday on the following Monday.
 * In Scotland, Boxing Day has been specified as an additional bank holiday since 1974, by royal proclamation under the Banking and Financial Dealings Act 1971.
 * In Singapore, Boxing Day was a public holiday for working people and students; when it fell on a Saturday or Sunday, there was a holiday on the following Monday. However, Boxing Day is no longer a public holiday.
 * In South Africa, 26 December is the Day of Goodwill, a public holiday.
 * In Trinidad and Tobago, Boxing Day is a public holiday.
 * In the UK outside Scotland, 26 December (unless it is a Sunday) has been a bank holiday since 1871. When 26 December falls on a Saturday, the associated public holiday is on the following Monday, 28 December. When 26 December falls on a Sunday, the public holiday is the following Tuesday 28 December, with the "substitute day" for Christmas Day being observed on the Monday. The same practice is observed in Canada.
 * In the British overseas territory of Bermuda, the costumed Gombey dancers perform throughout the mid-Atlantic island on Boxing Day, a tradition believed to date back to the 18th century, when slaves were permitted to gather at Christmas.
 * In Massachusetts, US, Governor William F. Weld declared in 1996 that every 26 December is Boxing Day, in response to the efforts of a coalition of British citizens to "transport the English tradition to the United States", but not an employee holiday. The holiday is otherwise not widely celebrated in the United States.

Shopping

In the United Kingdom, Canada, Australia, New Zealand, and Trinidad and Tobago, Boxing Day is primarily known as a shopping holiday. Boxing Day sales are common, and shops often allow dramatic price reductions. For many merchants, Boxing Day has become the day of the year with the greatest revenue. In the UK, it was estimated in 2009 that up to 12 million shoppers appeared at the sales (a rise of almost 20% compared to 2008, although this was also affected by the fact that the VAT was about to revert to 17.5% from 1 January, following the temporary reduction to 15%).

Many retailers open very early (typically 5 am or even earlier) and offer doorbuster deals and loss leaders to draw people to their stores. It is not uncommon for long queues to form early in the morning of 26 December, hours before the opening of shops holding the big sales, especially at big-box consumer electronics retailers. Many stores have a limited quantity of big draw or deeply discounted items. Because of the shoulder-to-shoulder crowds, many choose to stay at home and avoid the hectic shopping experience. Local media often covers the event, mentioning how early the shoppers began queuing up and showing videos of shoppers queuing and later leaving with their purchased items. Many retailers have implemented practices aimed at managing large numbers of shoppers. They may limit entrances, restrict the number of patrons in a store at a time, provide tickets to people at the head of the queue to guarantee them a hot ticket item, or canvass queued-up shoppers to inform them of inventory limitations.

In some areas of Canada, particularly in Atlantic Canada and parts of Northern Ontario, most retailers are prohibited from opening on Boxing Day, either by provincial law or by municipal bylaw, or by informal agreement among major retailers, to provide a day of relaxation following Christmas Day. In these areas, sales otherwise scheduled for 26 December are moved to the 27th. The city council of Greater Sudbury, Ontario, which was the largest city in Canada to maintain this restriction as of the early 2010s, formally repealed its store hours bylaw on 9 December 2014.

While Boxing Day is 26 December, many retailers run the sales for several days before or after 26 December, often up to New Year's Eve, branding it as "Boxing Week". Notably, in the recession of late 2008, a record number of retailers held early promotions because of the weak economy. In 2009, many retailers with both online and High Street stores launched their online sales on Christmas Eve and their High Street sales on Boxing Day.

Comparisons to Black Friday
In terms of seasonal or holiday shopping traditions, Boxing Day sales have been compared to the U.S. phenomenon of Black Friday sales –  Black Friday being the Friday following the American Thanksgiving holiday in late November. In the late 2000s, when the Canadian and United States dollars were near parity, Canadian retailers began to hold Black Friday promotions to attract consumers who would otherwise travel across the border to visit United States stores. This may have been a contributory factor, since 2013, in a relative decline of traditional Canadian Boxing Day sales, when compared to sales on Black Friday.

The traditional Boxing Day sales in the United Kingdom were never as large an event as the Black Friday sales are in the United States. However, many British retailers began to see an opportunity to import the Black Friday tradition into the UK, not to replace Boxing Day sales, but as an addition to their overall seasonal promotions. However, Black Friday and Boxing Day are close enough together that spending on one sale was likely to affect spending on the other. Ultimately, the result was a marked decline in traditional Boxing Day sales in the UK. The change was initially facilitated, although not necessarily by design, by the fact that many retailers had American ownership, such as Amazon. This phenomenon was furthered by a general decline in traditional high-street shopping and a growing online marketplace, which is more international by nature. This led, in 2015, to greater November retail sales in the UK than in December for the first time. In 2019, a retail analysis firm estimated that there was a 9.8% drop in British store traffic on Boxing Day in comparison to 2018 (the largest year-over-year drop since 2010), citing several factors, such as the weather, the increased prominence of online shopping, uncertainties in the wake of the general election, and the growing prominence of Black Friday sales.

Boxing Day sales are not a prominent tradition in the United States, although many retailers often begin after-Christmas sales that day. It is typically the earliest starting day after Christmas for people to return unwanted gifts for exchanges or refunds and to redeem gift cards.

Sport

In the United Kingdom, it is traditional for all top-tier football leagues in England, Scotland, and Northern Ireland—the Premier League, the Scottish Premiership, and the NIFL Premiership—and the lower ones, as well as the rugby leagues, to hold a full programme of football matches on Boxing Day. Originally, matches on Boxing Day were played against local rivals so that teams and their fans would not have to travel long distances to away games on the day after Christmas. Before the formation of leagues, several traditional rugby union fixtures took place on Boxing Day each year, notably Llanelli v London Welsh and Leicester v The Barbarians. The 2022 Premier League Boxing Day fixtures saw the return of domestic top flight football for the 2022–23 Premier League season, following the six-week break for to the 2022 FIFA World Cup.

In Italy, Boxing Day football was played for the first time in the 2018–19 Serie A season. The experiment was successful, with Italian stadiums 69% full on average – more than any other match day in December 2018.

In rugby league, festive fixtures were a staple of the traditional winter season. Since the transition to a summer season in the 1990s, no formal fixtures are now arranged on Boxing Day but some clubs, such as Wakefield Trinity, arrange a traditional local derby friendly fixture instead.

Since 1980, the Australian cricket team has traditionally opened an annual test match on Boxing Day at the Melbourne Cricket Ground. While several test matches had occasionally been held at the MCG around Boxing Day, it was not until 1980 that the concept was formalized by the Australian Cricket Board. The Sydney to Hobart Yacht Race is also traditionally held on Boxing Day.

In horse racing, there is the King George VI Chase at Kempton Park Racecourse in Surrey, England. It is the second most prestigious chase in Britain, after the Cheltenham Gold Cup. In addition to the prestigious race at Kempton, in Britain, it is usually the day with the most racing meetings of the year, with eight in 2016, in addition to three more in Ireland. In Barbados, the final day of horse racing is held on Boxing Day at The Historic Garrison Savannah, a UNESCO world heritage site. This tradition has been going on for decades in this former British colony.

Boxing Day is one of the main days in the hunting calendar for hunts in the UK and US, with most hunts (both mounted foxhound or harrier packs and foot packs of beagles or bassets) holding meets, often in town or village centres.

Several ice hockey contests are associated with the day. The IIHF World Junior Championship typically begins on 26 December, while the Spengler Cup also begins on 23 December in Davos, Switzerland; the Spengler Cup competition includes HC Davos, Team Canada, and other top European hockey teams. The National Hockey League traditionally had close to a full slate of games (10 were played in 2011 ), following the league-wide days off given for Christmas Eve and Christmas Day. However, the 2013 collective bargaining agreement (which followed a lock-out) extended the league mandate of Christmas Eve and Christmas Day off to include Boxing Day, except when it falls on a Saturday, in which case the league can choose to make 23 December a league-wide off day instead for that year. In Sweden, the related sport of bandy is also associated with the day, with Saint Stephen's Day bandy games having become an established tradition.

In some African Commonwealth nations, particularly Ghana, Uganda, Malawi, Zambia, and Tanzania, professional boxing contests are held on Boxing Day. This practice has also been followed for decades in Guyana and Italy.

Food
In the UK it is common to eat leftovers from the previous day's Christmas dinner, with turkey often being used in a Boxing Day sandwich or curry.

Boxing Day Tsunami
The 2004 Indian Ocean earthquake and tsunami occurred on 26 December and thus has been referred to as "the Boxing Day Tsunami".

title: Boxing Day


text:
Talk:Boxing Day

Incorrect information, and an overly zealous approach to blocking
The article is currently blocked due to "persistent vandalism", so I can't correct an error. The "persistent vandalism" consisted of one formatting mistake by an IP editor, followed by a single instance of silliness by another IP editor; this is hardly persistent! Anyway, someone has removed the statement that Boxing Day always falls on 26th December, even when this date is a Saturday or Sunday, claiming this to be untrue. In fact, it is true. Boxing Day is always 26th December.

I'm sorry but you are wrong. It was, and possibly still is, widely held that Boxing Day cannot fall on a Sunday, but it is not universally held. I can remember one year (probably in the 1976, maybe 1971) when the Radio Times (then a BBC only TV/Radio listings magazine) labelled Monday 27th as Boxing Day, whereas the TV Times (non-BBC TV listings) labelled Sunday 26th as Boxing Day.

Whether this belief arose because 27 December would be a Bank Holiday if 26 December is a Sunday I cannot say. The Acts of Parliament make no use of the term 'Boxing Day' and I think Wikipedia would be well advised to do the same thing.

See this, for example (and there are many more):. Could someone please correct the article. I'd do it myself - but I'm not allowed. Thanks. 31.52.165.66 (talk) 00:02, 27 December 2019 (UTC)
 * The gov.UK list of bank holidays shows Boxing Day on Dec 28 (a Monday) in 2020. Schazjmd   (talk)  00:11, 27 December 2019 (UTC)
 * Maybe I'm misunderstanding your point. You're saying Boxing Day = December 26th no matter what, but the bank holiday associated it with it might be on the 27th or 28th? Schazjmd   (talk)  00:14, 27 December 2019 (UTC)
 * And now I'm reading the article and it's saying exactly what I just said. I'm still not clear what you want added, and even the article you linked to says Boxing Day is a bank holiday. If Boxing Day falls on a Saturday, the following Monday is a bank holiday. Schazjmd   (talk)  00:17, 27 December 2019 (UTC)
 * I think the confusion is due to the need to differentiate between Boxing Day and its associated bank holiday; they are not quite the same thing. Perhaps to clarify, consider Christmas Day. If 25th Dec falls on a Saturday it is still Christmas Day, but the related bank holiday is moved to Monday, 27th (in which case the Boxing Day bank holiday is moved to Tuesday, 28th, but Boxing Day itself is still Sunday, 26th). At least, this is how it is in the UK. This is the edit I was concerned about . That said, the wording of the deleted sentence could be slightly improved. 31.52.165.66 (talk) 00:23, 27 December 2019 (UTC)
 * I think the article covers that: Boxing Day is on 26 December, although the attached bank holiday or public holiday may take place either on that day or two days later. Schazjmd   (talk)  00:32, 27 December 2019 (UTC)
 * In the section Status by country this is the problem statement:  When 26 December falls on a Saturday, Boxing Day is moved to the following Monday. If 26 December falls on a Sunday, the substitute public holiday is the following Tuesday. Here's what it should read: If Boxing Day falls on a Saturday or Sunday the associated public holiday is moved to the following Monday or Tuesday. 31.52.160.202 (talk) 11:14, 27 December 2019 (UTC)
 * I don't think that change (being less specific) is an improvement. I changed this: When 26 December falls on a Saturday, Boxing Day the public holiday is on the following Monday. Schazjmd   (talk)  18:42, 27 December 2019 (UTC)

Purely as an observation re the above: I was born in the UK in 1954. When I was a child Boxing Day was always the first weekday after Christmas. I'm not talking about holidays here, just 'Boxing Day' proper. If the 25th was a Saturday the 26th was called 'Christmas Sunday' and the 27th was Boxing Day (see also entry for Christmas Sunday which is not one I have worked on). If the 25th was a Friday then Saturday was 'Christmas Saturday', the 27th was 'Christmas Sunday' and the 28th was Boxing Day. At all other times Boxing Day was the 26th. I can't point to published authorities for this, only the fact that I lived it. From the references to the TV guides in the 1970s above it would appear this usage was fading by then. Given that this is only personal experience I'm not proposing changes to the main text.JordiYiman (talk) 00:05, 20 January 2022 (UTC)

Boxing Day (Day after Christmas) Celebration
Actually the next day after Christmas is a celebrated day in all Christianity as the day of Love. There are many languages where Love is a name for a female and there is a saint Love in the Christian Religion. The name Boxing Day is just an Anglo-Saxon custom. It is a holiday in all Christian countries and if it falls in a weekend the holiday is moved to the next working day. — Preceding unsigned comment added by 79.64.120.7 (talk) 12:46, 14 January 2020 (UTC)

This is utter nonsense. There is no saint Love and "the day of love" is not celebrated in all Christianity. Where's the source?

Bank Holidays | Boxing Day on a Sunday
The whole section on Bank Holidays is superfluous here at least as far as the UK is concerned. UK legislation does not use the term Boxing Day, so to say that Boxing Day is a Bank Holiday is at worst contentious and at best meaningless. Bank Holidays are covered in other articles so there is no need to go into the details of them here.

The author who is asserting that Boxing Day is always December 26 is wrong, and even the article itself contradicts him/her, citing the first recorded reference to Boxing Day explaining that it is the first weekday after Christmas Day - a very clear assertion that it does not fall on a Sunday. When I corrected the intro to the article in this vein, and admin asked me to give a source (so I just used the same source as in the "date" section). Why is no source required for the assertion that Boxing Day always falls on 26/12? That workers received their Christmas Box on a Sunday is unthinkable, so if that is origin then the assertion is wrong.

As there is no legal definition of Boxing Day, and it is not a religious holiday, its meaning all comes down to custom and tradition. The only facts appear to be that some people hold that it always falls on 26 December, whereas others hold that it does not fall on a Sunday. — Preceding unsigned comment added by Cardinal 1962 (talk • contribs) 09:45, 10 April 2020 (UTC)

Effectively the older tradition has been largely superseded and there are clashes because rules people used to be taught are no longer abided by. I remember a mess in 1993 (the first Sunday 26th December in over a decade) when even the Radio and TV Times disagreed. Increasingly a lot of post Christmas activities happen or start on the 26th (not that long ago things often waited until the first full weekday) and are invariably referred to as "Boxing Day" every year - next year the "Boxing Day sales" will start on Sunday, there will be "Boxing Day football" and so forth and they will all be using "Boxing Day" to mean that very day. However some older traditional events will not start/happen on Sunday 26th - for instance the "Boxing Day hunts" will not happen until the next day. Timrollpickering (talk) 14:22, 23 July 2020 (UTC)

In 1993 (and before) Radio Times called Sunday 26 December "Christmas Sunday" and the next day Boxing Day https://video-collection-international.fandom.com/wiki/Radio_Times:_18th_December_1993_to_1st_January_1994#Monday_.28December_27.2C_1993.29_.28Boxing_Day.29

But in 1999 they changed their minds - https://video-collection-international.fandom.com/wiki/Radio_Times:_18th_to_31st_December_1999#Sunday_.28December_26.2C_1999.29_.28Boxing_Day.29

 * It seems there was internal debate at the Radio Times and come 1999 the Sunday 26 side won. But contrary to what the article suggests I don't think it was the 1994 Sunday Trading Act that changed things. Instead it seems the BBC was a late hold out with a big internal debate in 1993 and then come 1999 they just conformed to what everyone else was doing. Others had gone for Sunday 26 in 1993 including the TV Times and Whitaker's Almanack. Football seems to have been split between the 26 and 27. The Liverpool Echo suggests the sales were on the Sunday - remember that the old Sunday closing laws were widely flouted especially if the fines would be exceeded by the potential profits. Timrollpickering (talk) 12:57, 23 January 2024 (UTC)

Irony
It seems ironic that there isn't more boxing on Boxing Day. Is there anywhere else in the world that has traditions like Africa mentioned in the article? Royal Autumn Crest (talk) 02:21, 11 December 2021 (UTC)

Saint Stephen's day
In Romania 27the of December is Saint Stepgen's Day and in Hungary as well as far as I am aware.. 2A04:241E:2500:E80:40D6:AB7C:BCEF:FF27 (talk) 19:36, 28 December 2021 (UTC)

Boxing day
Tell.me the meaning 105.112.30.120 (talk) 23:24, 25 December 2022 (UTC)

toronto
any pictures from elsewhere? 3MRB1 (talk) 07:02, 9 February 2024 (UTC)

It is fuckin great
It is!! 187.188.166.213 (talk) 14:36, 14 May 2024 (UTC)

title: Talk:Boxing Day