In
mathematics, the binomial coefficients are the positive
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s that occur as
coefficients in the
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 into a sum involving terms of the form , where the ...
. Commonly, a binomial coefficient is indexed by a pair of integers and is written
It is the coefficient of the term in the
polynomial expansion
In mathematics, an expansion of a product of sums expresses it as a sum of products by using the fact that multiplication distributes over addition. Expansion of a polynomial expression can be obtained by repeatedly replacing subexpressions tha ...
of the
binomial power
Power most often refers to:
* Power (physics), meaning "rate of doing work"
** Engine power, the power put out by an engine
** Electric power
* Power (social and political), the ability to influence people or events
** Abusive power
Power may a ...
; this coefficient can be computed by the multiplicative formula
:
which using
factorial notation can be compactly expressed as
:
For example, the fourth power of is
:
and the binomial coefficient
is the coefficient of the term.
Arranging the numbers
in successive rows for
gives a triangular array called
Pascal's triangle
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although o ...
, satisfying the
recurrence relation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
:
The binomial coefficients occur in many areas of mathematics, and especially in
combinatorics. The symbol
is usually read as " choose " because there are
ways to choose an (unordered) subset of elements from a fixed set of elements. For example, there are
ways to choose 2 elements from
namely
and
The binomial coefficients can be generalized to
for any complex number and integer , and many of their properties continue to hold in this more general form.
History and notation
Andreas von Ettingshausen introduced the notation
in 1826, although the numbers were known centuries earlier (see
Pascal's triangle
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although o ...
). The earliest known detailed discussion of binomial coefficients is in a tenth-century commentary, by
Halayudha
Halayudha (Sanskrit: हलायुध) was a 10th-century Indian mathematician who wrote the ',Maurice Winternitz, ''History of Indian Literature'', Vol. III a commentary on Pingala's ''Chandaḥśāstra''. The latter contains a clear descri ...
, on an ancient
Sanskrit
Sanskrit (; attributively , ; nominally , , ) is a classical language belonging to the Indo-Aryan branch of the Indo-European languages. It arose in South Asia after its predecessor languages had diffused there from the northwest in the late ...
text,
Pingala
Acharya Pingala ('; c. 3rd2nd century BCE) was an ancient Indian poet and mathematician, and the author of the ' (also called the ''Pingala-sutras''), the earliest known treatise on Sanskrit prosody.
The ' is a work of eight chapters in the la ...
's ''Chandaḥśāstra''. The second earliest description of binomial coefficients is given by
Al-Karaji. In about 1150, the Indian mathematician
Bhaskaracharya gave an exposition of binomial coefficients in his book ''
Līlāvatī''.
Alternative notations include , , , , , and in all of which the stands for ''
combination
In mathematics, a combination is a selection of items from a set that has distinct members, such that the order of selection does not matter (unlike permutations). For example, given three fruits, say an apple, an orange and a pear, there are th ...
s'' or ''choices''. Many calculators use variants of the because they can represent it on a single-line display. In this form the binomial coefficients are easily compared to
-permutations of , written as , etc.
Definition and interpretations
For
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''cardinal ...
s (taken to include 0) ''n'' and ''k'', the binomial coefficient
can be defined as the
coefficient of the
monomial ''X''
''k'' in the expansion of . The same coefficient also occurs (if ) in the
binomial formula
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 into a sum involving terms of the form , where the ...
(valid for any elements ''x'', ''y'' of a
commutative ring),
which explains the name "binomial coefficient".
Another occurrence of this number is in combinatorics, where it gives the number of ways, disregarding order, that ''k'' objects can be chosen from among ''n'' objects; more formally, the number of ''k''-element subsets (or ''k''-
combination
In mathematics, a combination is a selection of items from a set that has distinct members, such that the order of selection does not matter (unlike permutations). For example, given three fruits, say an apple, an orange and a pear, there are th ...
s) of an ''n''-element set. This number can be seen as equal to the one of the first definition, independently of any of the formulas below to compute it: if in each of the ''n'' factors of the power one temporarily labels the term ''X'' with an index ''i'' (running from 1 to ''n''), then each subset of ''k'' indices gives after expansion a contribution ''X''
''k'', and the coefficient of that monomial in the result will be the number of such subsets. This shows in particular that
is a natural number for any natural numbers ''n'' and ''k''. There are many other combinatorial interpretations of binomial coefficients (counting problems for which the answer is given by a binomial coefficient expression), for instance the number of words formed of ''n''
bit
The bit is the most basic unit of information in computing and digital communications. The name is a portmanteau of binary digit. The bit represents a logical state with one of two possible values. These values are most commonly represente ...
s (digits 0 or 1) whose sum is ''k'' is given by
, while the number of ways to write
where every ''a''
''i'' is a nonnegative integer is given by
. Most of these interpretations are easily seen to be equivalent to counting ''k''-combinations.
Computing the value of binomial coefficients
Several methods exist to compute the value of
without actually expanding a binomial power or counting ''k''-combinations.
Recursive formula
One method uses the
recursive
Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics ...
, purely additive formula
for all integers
such that
with initial/boundary values
for all integers
The formula follows from considering the set and counting separately (a) the ''k''-element groupings that include a particular set element, say "''i''", in every group (since "''i''" is already chosen to fill one spot in every group, we need only choose from the remaining ) and (b) all the ''k''-groupings that don't include "''i''"; this enumerates all the possible ''k''-combinations of ''n'' elements. It also follows from tracing the contributions to ''X''
''k'' in . As there is zero or in , one might extend the definition beyond the above boundaries to include
when either or . This recursive formula then allows the construction of
Pascal's triangle
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although o ...
, surrounded by white spaces where the zeros, or the trivial coefficients, would be.
Multiplicative formula
A more efficient method to compute individual binomial coefficients is given by the formula
where the numerator of the first fraction
is expressed as a
falling factorial power
In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial
:\begin
(x)_n = x^\underline &= \overbrace^ \\
&= \prod_^n(x-k+1) = \prod_^(x-k) \,.
\e ...
.
This formula is easiest to understand for the combinatorial interpretation of binomial coefficients.
The numerator gives the number of ways to select a sequence of ''k'' distinct objects, retaining the order of selection, from a set of ''n'' objects. The denominator counts the number of distinct sequences that define the same ''k''-combination when order is disregarded.
Due to the symmetry of the binomial coefficient with regard to ''k'' and , calculation may be optimised by setting the upper limit of the product above to the smaller of ''k'' and .
Factorial formula
Finally, though computationally unsuitable, there is the compact form, often used in proofs and derivations, which makes repeated use of the familiar
factorial function:
where ''n''! denotes the factorial of ''n''. This formula follows from the multiplicative formula above by multiplying numerator and denominator by ; as a consequence it involves many factors common to numerator and denominator. It is less practical for explicit computation (in the case that ''k'' is small and ''n'' is large) unless common factors are first cancelled (in particular since factorial values grow very rapidly). The formula does exhibit a symmetry that is less evident from the multiplicative formula (though it is from the definitions)
which leads to a more efficient multiplicative computational routine. Using the
falling factorial notation,
Generalization and connection to the binomial series
The multiplicative formula allows the definition of binomial coefficients to be extended by replacing ''n'' by an arbitrary number ''α'' (negative, real, complex) or even an element of any
commutative ring in which all positive integers are invertible:
With this definition one has a generalization of the binomial formula (with one of the variables set to 1), which justifies still calling the
binomial coefficients:
This formula is valid for all complex numbers ''α'' and ''X'' with , ''X'', < 1. It can also be interpreted as an identity of
formal power series in ''X'', where it actually can serve as definition of arbitrary powers of power series with constant coefficient equal to 1; the point is that with this definition all identities hold that one expects for
exponentiation
Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to r ...
, notably
If ''α'' is a nonnegative integer ''n'', then all terms with are zero, and the infinite series becomes a finite sum, thereby recovering the binomial formula. However, for other values of ''α'', including negative integers and rational numbers, the series is really infinite.
Pascal's triangle
Pascal's rule
In mathematics, Pascal's rule (or Pascal's formula) is a combinatorial identity about binomial coefficients. It states that for positive natural numbers ''n'' and ''k'',
+ = ,
where \tbinom is a binomial coefficient; one interpretation of ...
is the important
recurrence relation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
which can be used to prove by
mathematical induction
Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ... all hold. Informal metaphors help ...
that
is a natural number for all integer ''n'' ≥ 0 and all integer ''k'', a fact that is not immediately obvious from
formula (1). To the left and right of Pascal's triangle, the entries (shown as blanks) are all zero.
Pascal's rule also gives rise to
Pascal's triangle
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although o ...
:
Row number contains the numbers
for . It is constructed by first placing 1s in the outermost positions, and then filling each inner position with the sum of the two numbers directly above. This method allows the quick calculation of binomial coefficients without the need for fractions or multiplications. For instance, by looking at row number 5 of the triangle, one can quickly read off that
:
Combinatorics and statistics
Binomial coefficients are of importance in
combinatorics, because they provide ready formulas for certain frequent counting problems:
* There are
ways to choose ''k'' elements from a set of ''n'' elements. See
Combination
In mathematics, a combination is a selection of items from a set that has distinct members, such that the order of selection does not matter (unlike permutations). For example, given three fruits, say an apple, an orange and a pear, there are th ...
.
* There are
ways to choose ''k'' elements from a set of ''n'' elements if repetitions are allowed. See
Multiset
In mathematics, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements. The number of instances given for each element is called the multiplicity of that e ...
.
* There are
strings containing ''k'' ones and ''n'' zeros.
* There are
strings consisting of ''k'' ones and ''n'' zeros such that no two ones are adjacent.
* The
Catalan number
In combinatorial mathematics, the Catalan numbers are a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. They are named after the French-Belgian mathematician Eugène Charles Ca ...
s are
* The
binomial distribution in
statistics is
Binomial coefficients as polynomials
For any nonnegative integer ''k'', the expression
can be simplified and defined as a polynomial divided by :
:
this presents a
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
in ''t'' with
rational
Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ...
coefficients.
As such, it can be evaluated at any real or complex number ''t'' to define binomial coefficients with such first arguments. These "generalized binomial coefficients" appear in
Newton's generalized 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 into a sum involving terms of the form , where the ...
.
For each ''k'', the polynomial
can be characterized as the unique degree ''k'' polynomial satisfying and .
Its coefficients are expressible in terms of
Stirling numbers of the first kind
In mathematics, especially in combinatorics, Stirling numbers of the first kind arise in the study of permutations. In particular, the Stirling numbers of the first kind count permutations according to their number of cycles (counting fixed poin ...
:
:
The
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
of
can be calculated by
logarithmic differentiation
In calculus, logarithmic differentiation or differentiation by taking logarithms is a method used to differentiate functions by employing the logarithmic derivative of a function ''f'',
:(\ln f)' = \frac \quad \implies \quad f' = f \cdot (\ ...
:
:
This can cause a problem when evaluated at integers from
to
, but using identities below we can compute the derivative as:
:
Binomial coefficients as a basis for the space of polynomials
Over any
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
of
characteristic 0
In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive i ...
(that is, any field that contains the
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rat ...
s), each polynomial ''p''(''t'') of degree at most ''d'' is uniquely expressible as a linear combination
of binomial coefficients. The coefficient ''a''
''k'' is the
''k''th difference of the sequence ''p''(0), ''p''(1), ..., ''p''(''k''). Explicitly,
Integer-valued polynomials
Each polynomial
is
integer-valued: it has an integer value at all integer inputs
. (One way to prove this is by induction on ''k'', using
Pascal's identity
In mathematics, Pascal's rule (or Pascal's formula) is a combinatorial identity about binomial coefficients. It states that for positive natural numbers ''n'' and ''k'',
+ = ,
where \tbinom is a binomial coefficient; one interpretation of ...
.) Therefore, any integer linear combination of binomial coefficient polynomials is integer-valued too. Conversely, () shows that any integer-valued polynomial is an integer linear combination of these binomial coefficient polynomials. More generally, for any subring ''R'' of a characteristic 0 field ''K'', a polynomial in ''K''
't''takes values in ''R'' at all integers if and only if it is an ''R''-linear combination of binomial coefficient polynomials.
Example
The integer-valued polynomial can be rewritten as
:
Identities involving binomial coefficients
The factorial formula facilitates relating nearby binomial coefficients. For instance, if ''k'' is a positive integer and ''n'' is arbitrary, then
and, with a little more work,
:
We can also get
:
Moreover, the following may be useful:
:
For constant ''n'', we have the following recurrence:
:
To sum up, we have
:
Sums of the binomial coefficients
The formula
says the elements in the th row of Pascal's triangle always add up to 2 raised to the th power. This is obtained from the binomial theorem () by setting ''x'' = 1 and ''y'' = 1. The formula also has a natural combinatorial interpretation: the left side sums the number of subsets of of sizes ''k'' = 0, 1, ..., ''n'', giving the total number of subsets. (That is, the left side counts the
power set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is post ...
of .) However, these subsets can also be generated by successively choosing or excluding each element 1, ..., ''n''; the ''n'' independent binary choices (bit-strings) allow a total of
choices. The left and right sides are two ways to count the same collection of subsets, so they are equal.
The formulas
and
:
follow from the binomial theorem after
differentiating with respect to (twice for the latter) and then substituting .
The
Chu–Vandermonde identity, which holds for any complex values ''m'' and ''n'' and any non-negative integer ''k'', is
and can be found by examination of the coefficient of
in the expansion of using equation (). When , equation () reduces to equation (). In the special case , using (), the expansion () becomes (as seen in Pascal's triangle at right)
where the term on the right side is a
central binomial coefficient.
Another form of the Chu–Vandermonde identity, which applies for any integers ''j'', ''k'', and ''n'' satisfying , is
The proof is similar, but uses the binomial series expansion () with negative integer exponents.
When , equation () gives the
hockey-stick identity
In combinatorial mathematics, the identity
: \sum^n_= \qquad \text n,r\in\mathbb, \quad n\geq r
or equivalently, the mirror-image by the substitution j\to i-r:
: \sum^_=\sum^_= \qquad \text n,r\in\mathbb, \quad n\geq r
is known as the hockey ...
:
and its relative
:
Let ''F''(''n'') denote the ''n''-th
Fibonacci number
In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...
.
Then
:
This can be proved by
induction
Induction, Inducible or Inductive may refer to:
Biology and medicine
* Labor induction (birth/pregnancy)
* Induction chemotherapy, in medicine
* Induced stem cells, stem cells derived from somatic, reproductive, pluripotent or other cell t ...
using () or by
Zeckendorf's representation. A combinatorial proof is given below.
Multisections of sums
For integers ''s'' and ''t'' such that
series multisection gives the following identity for the sum of binomial coefficients:
:
For small , these series have particularly nice forms; for example,
:
:
:
:
:
:
:
Partial sums
Although there is no
closed formula for
partial sum
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
s
:
of binomial coefficients, one can again use () and induction to show that for ,
:
with special case
:
for ''n'' > 0. This latter result is also a special case of the result from the theory of
finite differences
A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the ...
that for any polynomial ''P''(''x'') of degree less than ''n'',
:
Differentiating () ''k'' times and setting ''x'' = −1 yields this for
,
when 0 ≤ ''k'' < ''n'',
and the general case follows by taking linear combinations of these.
When ''P''(''x'') is of degree less than or equal to ''n'',
where
is the coefficient of degree ''n'' in ''P''(''x'').
More generally for (),
:
where ''m'' and ''d'' are complex numbers. This follows immediately applying () to the polynomial instead of , and observing that still has degree less than or equal to ''n'', and that its coefficient of degree ''n'' is ''d
na
n''.
The
series
Series may refer to:
People with the name
* Caroline Series (born 1951), English mathematician, daughter of George Series
* George Series (1920–1995), English physicist
Arts, entertainment, and media
Music
* Series, the ordered sets used in ...
is convergent for ''k'' ≥ 2. This formula is used in the analysis of the
German tank problem
In the statistical theory of estimation, the German tank problem consists of estimating the maximum of a discrete uniform distribution from sampling without replacement. In simple terms, suppose there exists an unknown number of items which are s ...
. It follows from
which is proved by
induction
Induction, Inducible or Inductive may refer to:
Biology and medicine
* Labor induction (birth/pregnancy)
* Induction chemotherapy, in medicine
* Induced stem cells, stem cells derived from somatic, reproductive, pluripotent or other cell t ...
on ''M''.
Identities with combinatorial proofs
Many identities involving binomial coefficients can be proved by
combinatorial means. For example, for nonnegative integers
, the identity
:
(which reduces to () when ''q'' = 1) can be given a
double counting proof, as follows. The left side counts the number of ways of selecting a subset of
'n''= with at least ''q'' elements, and marking ''q'' elements among those selected. The right side counts the same thing, because there are
ways of choosing a set of ''q'' elements to mark, and
to choose which of the remaining elements of
'n''also belong to the subset.
In Pascal's identity
:
both sides count the number of ''k''-element subsets of
'n'' the two terms on the right side group them into those that contain element ''n'' and those that do not.
The identity () also has a combinatorial proof. The identity reads
:
Suppose you have
empty squares arranged in a row and you want to mark (select) ''n'' of them. There are
ways to do this. On the other hand, you may select your ''n'' squares by selecting ''k'' squares from among the first ''n'' and
squares from the remaining ''n'' squares; any ''k'' from 0 to ''n'' will work. This gives
:
Now apply () to get the result.
If one denotes by the sequence of
Fibonacci number
In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...
s, indexed so that , then the identity
has the following combinatorial proof. One may show by
induction
Induction, Inducible or Inductive may refer to:
Biology and medicine
* Labor induction (birth/pregnancy)
* Induction chemotherapy, in medicine
* Induced stem cells, stem cells derived from somatic, reproductive, pluripotent or other cell t ...
that counts the number of ways that a strip of squares may be covered by and tiles. On the other hand, if such a tiling uses exactly of the tiles, then it uses of the tiles, and so uses tiles total. There are
ways to order these tiles, and so summing this coefficient over all possible values of gives the identity.
Sum of coefficients row
The number of ''k''-
combinations
In mathematics, a combination is a selection of items from a set that has distinct members, such that the order of selection does not matter (unlike permutations). For example, given three fruits, say an apple, an orange and a pear, there are th ...
for all ''k'',
, is the sum of the ''n''th row (counting from 0) of the binomial coefficients. These combinations are enumerated by the 1 digits of the set of
base 2 numbers counting from 0 to
, where each digit position is an item from the set of ''n''.
Dixon's identity
Dixon's identity is
:
or, more generally,
:
where ''a'', ''b'', and ''c'' are non-negative integers.
Continuous identities
Certain trigonometric integrals have values expressible in terms of binomial coefficients: For any
:
:
:
These can be proved by using
Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that fo ...
to convert
trigonometric functions
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in al ...
to complex exponentials, expanding using the binomial theorem, and integrating term by term.
Congruences
If ''n'' is prime, then
for every ''k'' with
More generally, this remains true if ''n'' is any number and ''k'' is such that all the numbers between 1 and ''k'' are coprime to ''n''.
Indeed, we have
:
Generating functions
Ordinary generating functions
For a fixed , the
ordinary generating function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary ser ...
of the sequence
is
:
For a fixed , the ordinary generating function of the sequence
is
:
The
bivariate generating function of the binomial coefficients is
:
A symmetric bivariate generating function of the binomial coefficients is
:
which is the same as the previous generating function after the substitution
.
Exponential generating function
A symmetric
exponential bivariate generating function of the binomial coefficients is:
:
Divisibility properties
In 1852,
Kummer proved that if ''m'' and ''n'' are nonnegative integers and ''p'' is a prime number, then the largest power of ''p'' dividing
equals ''p''
''c'', where ''c'' is the number of carries when ''m'' and ''n'' are added in base ''p''.
Equivalently, the exponent of a prime ''p'' in
equals the number of nonnegative integers ''j'' such that the
fractional part of ''k''/''p''
''j'' is greater than the fractional part of ''n''/''p''
''j''. It can be deduced from this that
is divisible by ''n''/
gcd(''n'',''k''). In particular therefore it follows that ''p'' divides
for all positive integers ''r'' and ''s'' such that . However this is not true of higher powers of ''p'': for example 9 does not divide
.
A somewhat surprising result by
David Singmaster
David Breyer Singmaster (born 1938) is an emeritus professor of mathematics at London South Bank University, England. A self-described metagrobologist, he has a huge personal collection of mechanical puzzles and books of brain teasers. He is mo ...
(1974) is that any integer divides
almost all binomial coefficients. More precisely, fix an integer ''d'' and let ''f''(''N'') denote the number of binomial coefficients
with ''n'' < ''N'' such that ''d'' divides
. Then
:
Since the number of binomial coefficients
with ''n'' < ''N'' is ''N''(''N'' + 1) / 2, this implies that the density of binomial coefficients divisible by ''d'' goes to 1.
Binomial coefficients have divisibility properties related to least common multiples of consecutive integers. For example:
divides
.
is a multiple of
.
Another fact:
An integer is prime if and only if
all the intermediate binomial coefficients
:
are divisible by ''n''.
Proof:
When ''p'' is prime, ''p'' divides
:
for all
because
is a natural number and ''p'' divides the numerator but not the denominator.
When ''n'' is composite, let ''p'' be the smallest prime factor of ''n'' and let . Then and
:
otherwise the numerator has to be divisible by , this can only be the case when is divisible by ''p''. But ''n'' is divisible by ''p'', so ''p'' does not divide and because ''p'' is prime, we know that ''p'' does not divide and so the numerator cannot be divisible by ''n''.
Bounds and asymptotic formulas
The following bounds for
hold for all values of ''n'' and ''k'' such that :
The first inequality follows from the fact that
and each of these
terms in this product is
. A similar argument can be made to show the second inequality. The final strict inequality is equivalent to
, that is clear since the RHS is a term of the exponential series
.
From the divisibility properties we can infer that
where both equalities can be achieved.
The following bounds are useful in information theory:
where
is the
binary entropy function
In information theory, the binary entropy function, denoted \operatorname H(p) or \operatorname H_\text(p), is defined as the entropy of a Bernoulli process with probability p of one of two values. It is a special case of \Eta(X), the entropy fun ...
. It can be further tightened to
for all
.
Both and large
Stirling's approximation
In mathematics, Stirling's approximation (or Stirling's formula) is an approximation for factorials. It is a good approximation, leading to accurate results even for small values of n. It is named after James Stirling, though a related but less p ...
yields the following approximation, valid when
both tend to infinity:
Because the inequality forms of Stirling's formula also bound the factorials, slight variants on the above asymptotic approximation give exact bounds.
In particular, when
is sufficiently large, one has
and
and, more generally, for and ,
If ''n'' is large and ''k'' is linear in ''n'', various precise asymptotic estimates exist for the binomial coefficient
. For example, if
then
where ''d'' = ''n'' − 2''k''.
much larger than
If is large and is (that is, if ), then
where again is the
little o notation
Big ''O'' notation is a mathematical notation that describes the asymptotic analysis, limiting behavior of a function (mathematics), function when the Argument of a function, argument tends towards a particular value or infinity. Big O is a memb ...
.
Sums of binomial coefficients
A simple and rough upper bound for the sum of binomial coefficients can be obtained using the
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 into a sum involving terms of the form , where the ...
:
More precise bounds are given by
valid for all integers
with
.
Generalized binomial coefficients
The
infinite product formula for the gamma function also gives an expression for binomial coefficients
which yields the asymptotic formulas
as
.
This asymptotic behaviour is contained in the approximation
as well. (Here
is the ''k''-th
harmonic number
In mathematics, the -th harmonic number is the sum of the reciprocals of the first natural numbers:
H_n= 1+\frac+\frac+\cdots+\frac =\sum_^n \frac.
Starting from , the sequence of harmonic numbers begins:
1, \frac, \frac, \frac, \frac, \dot ...
and
is the
Euler–Mascheroni constant
Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant usually denoted by the lowercase Greek letter gamma ().
It is defined as the limiting difference between the harmonic series and the natural l ...
.)
Further, the asymptotic formula
hold true, whenever
and
for some complex number
.
Generalizations
Generalization to multinomials
Binomial coefficients can be generalized to multinomial coefficients defined to be the number:
:
where
:
While the binomial coefficients represent the coefficients of (''x''+''y'')
''n'', the multinomial coefficients
represent the coefficients of the polynomial
:
The case ''r'' = 2 gives binomial coefficients:
:
The combinatorial interpretation of multinomial coefficients is distribution of ''n'' distinguishable elements over ''r'' (distinguishable) containers, each containing exactly ''k
i'' elements, where ''i'' is the index of the container.
Multinomial coefficients have many properties similar to those of binomial coefficients, for example the recurrence relation:
:
and symmetry:
:
where
is a
permutation of (1, 2, ..., ''r'').
Taylor series
Using
Stirling numbers of the first kind
In mathematics, especially in combinatorics, Stirling numbers of the first kind arise in the study of permutations. In particular, the Stirling numbers of the first kind count permutations according to their number of cycles (counting fixed poin ...
the
series expansion
In mathematics, a series expansion is an expansion of a function into a series, or infinite sum. It is a method for calculating a function that cannot be expressed by just elementary operators (addition, subtraction, multiplication and divisi ...
around any arbitrarily chosen point
is
:
Binomial coefficient with
The definition of the binomial coefficients can be extended to the case where
is real and
is integer.
In particular, the following identity holds for any non-negative integer
:
:
This shows up when expanding
into a power series using the Newton binomial series :
:
Products of binomial coefficients
One can express the product of two binomial coefficients as a linear combination of binomial coefficients:
:
where the connection coefficients are
multinomial coefficients
In mathematics, the multinomial theorem describes how to expand a power of a sum in terms of powers of the terms in that sum. It is the generalization of the binomial theorem from binomials to multinomials.
Theorem
For any positive integer ...
. In terms of labelled combinatorial objects, the connection coefficients represent the number of ways to assign labels to a pair of labelled combinatorial objects—of weight ''m'' and ''n'' respectively—that have had their first ''k'' labels identified, or glued together to get a new labelled combinatorial object of weight . (That is, to separate the labels into three portions to apply to the glued part, the unglued part of the first object, and the unglued part of the second object.) In this regard, binomial coefficients are to exponential generating series what
falling factorials are to ordinary generating series.
The product of all binomial coefficients in the ''n''th row of the Pascal triangle is given by the formula:
:
Partial fraction decomposition
The
partial fraction decomposition
In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as ...
of the reciprocal is given by
:
Newton's binomial series
Newton's binomial series, named after
Sir Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a " natural philosopher"), widely recognised as one of the g ...
, is a generalization of the binomial theorem to infinite series:
:
The identity can be obtained by showing that both sides satisfy the
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
.
The
radius of convergence
In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or \infty. When it is positive, the power series ...
of this series is 1. An alternative expression is
:
where the identity
:
is applied.
Multiset (rising) binomial coefficient
Binomial coefficients count subsets of prescribed size from a given set. A related combinatorial problem is to count
multiset
In mathematics, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements. The number of instances given for each element is called the multiplicity of that e ...
s of prescribed size with elements drawn from a given set, that is, to count the number of ways to select a certain number of elements from a given set with the possibility of selecting the same element repeatedly. The resulting numbers are called ''
multiset coefficients''; the number of ways to "multichoose" (i.e., choose with replacement) ''k'' items from an ''n'' element set is denoted
.
To avoid ambiguity and confusion with ''ns main denotation in this article,
let and .
Multiset coefficients may be expressed in terms of binomial coefficients by the rule
One possible alternative characterization of this identity is as follows:
We may define the
falling factorial as
and the corresponding rising factorial as
so, for example,
Then the binomial coefficients may be written as
while the corresponding multiset coefficient is defined by replacing the falling with the rising factorial:
Generalization to negative integers ''n''
For any ''n'',
:
In particular, binomial coefficients evaluated at negative integers ''n'' are given by signed multiset coefficients. In the special case
, this reduces to
For example, if ''n'' = −4 and ''k'' = 7, then ''r'' = 4 and ''f'' = 10:
:
Two real or complex valued arguments
The binomial coefficient is generalized to two real or complex valued arguments using the
gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
or
beta function
In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral
: \Beta(z_1,z_2) = \int_0^1 t^( ...
via
:
This definition inherits these following additional properties from
:
:
moreover,
:
The resulting function has been little-studied, apparently first being graphed in . Notably, many binomial identities fail:
but
for ''n'' positive (so
negative). The behavior is quite complex, and markedly different in various octants (that is, with respect to the ''x'' and ''y'' axes and the line
), with the behavior for negative ''x'' having singularities at negative integer values and a checkerboard of positive and negative regions:
* in the octant
it is a smoothly interpolated form of the usual binomial, with a ridge ("Pascal's ridge").
* in the octant
and in the quadrant
the function is close to zero.
* in the quadrant
the function is alternatingly very large positive and negative on the parallelograms with vertices
* in the octant
the behavior is again alternatingly very large positive and negative, but on a square grid.
* in the octant
it is close to zero, except for near the singularities.
Generalization to ''q''-series
The binomial coefficient has a
q-analog
In mathematics, a ''q''-analog of a theorem, identity or expression is a generalization involving a new parameter ''q'' that returns the original theorem, identity or expression in the limit as . Typically, mathematicians are interested in ''q' ...
generalization known as the
Gaussian binomial coefficient
In mathematics, the Gaussian binomial coefficients (also called Gaussian coefficients, Gaussian polynomials, or ''q''-binomial coefficients) are ''q''-analogs of the binomial coefficients. The Gaussian binomial coefficient, written as \binom n ...
.
Generalization to infinite cardinals
The definition of the binomial coefficient can be generalized to
infinite cardinals by defining:
:
where A is some set with
cardinality . One can show that the generalized binomial coefficient is well-defined, in the sense that no matter what set we choose to represent the
cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. T ...
,
will remain the same. For finite cardinals, this definition coincides with the standard definition of the binomial coefficient.
Assuming the
Axiom of Choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
, one can show that
for any infinite cardinal
.
In programming languages
The notation
is convenient in handwriting but inconvenient for
typewriter
A typewriter is a mechanical or electromechanical machine for typing characters. Typically, a typewriter has an array of keys, and each one causes a different single character to be produced on paper by striking an inked ribbon selectivel ...
s and
computer terminal
A computer terminal is an electronic or electromechanical hardware device that can be used for entering data into, and transcribing data from, a computer or a computing system. The teletype was an example of an early-day hard-copy terminal and ...
s. Many
programming language
A programming language is a system of notation for writing computer programs. Most programming languages are text-based formal languages, but they may also be graphical. They are a kind of computer language.
The description of a programming ...
s do not offer a standard
subroutine for computing the binomial coefficient, but for example both the
APL programming language
APL (named after the book ''A Programming Language'') is a programming language developed in the 1960s by Kenneth E. Iverson. Its central datatype is the multidimensional array. It uses a large range of special graphic symbols to represent mos ...
and the (related)
J programming language
The J programming language, developed in the early 1990s by Kenneth E. Iverson and Roger Hui, is an array programming language based primarily on APL (also by Iverson).
To avoid repeating the APL special-character problem, J uses only the basic ...
use the exclamation mark:
k ! n
. The binomial coefficient is implemented in
SciPy
SciPy (pronounced "sigh pie") is a free and open-source Python library used for scientific computing and technical computing.
SciPy contains modules for optimization, linear algebra, integration, interpolation, special functions, FFT, ...
as ''scipy.special.comb''.
Naive implementations of the factorial formula, such as the following snippet in
Python
Python may refer to:
Snakes
* Pythonidae, a family of nonvenomous snakes found in Africa, Asia, and Australia
** ''Python'' (genus), a genus of Pythonidae found in Africa and Asia
* Python (mythology), a mythical serpent
Computing
* Python (pro ...
:
from math import factorial
def binomial_coefficient(n: int, k: int) -> int:
return factorial(n) // (factorial(k) * factorial(n - k))
are very slow and are useless for calculating factorials of very high numbers (in languages such as
C or
Java
Java (; id, Jawa, ; jv, ꦗꦮ; su, ) is one of the Greater Sunda Islands in Indonesia. It is bordered by the Indian Ocean to the south and the Java Sea to the north. With a population of 151.6 million people, Java is the world's mos ...
they suffer from overflow errors because of this reason). A direct implementation of the multiplicative formula works well:
def binomial_coefficient(n: int, k: int) -> int:
if k < 0 or k > n:
return 0
if k 0 or k n:
return 1
k = min(k, n - k) # Take advantage of symmetry
c = 1
for i in range(k):
c = c * (n - i) // (i + 1)
return c
(In Python, range(k) produces a list from 0 to k−1.)
Pascal's rule
In mathematics, Pascal's rule (or Pascal's formula) is a combinatorial identity about binomial coefficients. It states that for positive natural numbers ''n'' and ''k'',
+ = ,
where \tbinom is a binomial coefficient; one interpretation of ...
provides a recursive definition which can also be implemented in Python, although it is less efficient:
def binomial_coefficient(n: int, k: int) -> int:
if k < 0 or k > n:
return 0
if k > n - k: # Take advantage of symmetry
k = n - k
if k 0 or n <= 1:
return 1
return binomial_coefficient(n - 1, k) + binomial_coefficient(n - 1, k - 1)
The example mentioned above can be also written in functional style. The following
Scheme example uses the recursive definition
:
Rational arithmetic can be easily avoided using integer division
:
The following implementation uses all these ideas
(define (binomial n k)
;; Helper function to compute C(n,k) via forward recursion
(define (binomial-iter n k i prev)
(if (>= i k)
prev
(binomial-iter n k (+ i 1) (/ (* (- n i) prev) (+ i 1)))))
;; Use symmetry property C(n,k)=C(n, n-k)
(if (< k (- n k))
(binomial-iter n k 0 1)
(binomial-iter n (- n k) 0 1)))
When computing
in a language with fixed-length integers, the multiplication by
may overflow even when the result would fit. The overflow can be avoided by dividing first and fixing the result using the remainder:
:
Implementation in the C language:
#include
unsigned long binomial(unsigned long n, unsigned long k)
Another way to compute the binomial coefficient when using large numbers is to recognize that
:
where
denotes the
natural logarithm of the
gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
at
. It is a special function that is easily computed and is standard in some programming languages such as using ''log_gamma'' in
Maxima, ''LogGamma'' in
Mathematica, ''gammaln'' in
MATLAB
MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementa ...
and Python's
SciPy
SciPy (pronounced "sigh pie") is a free and open-source Python library used for scientific computing and technical computing.
SciPy contains modules for optimization, linear algebra, integration, interpolation, special functions, FFT, ...
module, ''lngamma'' in
PARI/GP
PARI/GP is a computer algebra system with the main aim of facilitating number theory computations. Versions 2.1.0 and higher are distributed under the GNU General Public License. It runs on most common operating systems.
System overview
The ...
or ''lgamma'' in C,
R,
and
Julia
Julia is usually a feminine given name. It is a Latinate feminine form of the name Julio and Julius. (For further details on etymology, see the Wiktionary entry "Julius".) The given name ''Julia'' had been in use throughout Late Antiquity (e.g ...
. Roundoff error may cause the returned value to not be an integer.
See also
*
Binomial transform In combinatorics, the binomial transform is a sequence transformation (i.e., a transform of a sequence) that computes its forward differences. It is closely related to the Euler transform, which is the result of applying the binomial transform to t ...
*
Delannoy number
In mathematics, a Delannoy number D describes the number of paths from the southwest corner (0, 0) of a rectangular grid to the northeast corner (''m'', ''n''), using only single steps north, northeast, or east. The Delannoy numbers are named aft ...
*
Eulerian number
In combinatorics, the Eulerian number ''A''(''n'', ''m'') is the number of permutations of the numbers 1 to ''n'' in which exactly ''m'' elements are greater than the previous element (permutations with ''m'' "ascents"). They are the coefficients ...
*
Hypergeometric function
*
List of factorial and binomial topics {{Short description, none
This is a list of factorial and binomial topics in mathematics. See also binomial (disambiguation).
* Abel's binomial theorem
* Alternating factorial
*Antichain
*Beta function
*Bhargava factorial
*Binomial coefficient
**P ...
*
Macaulay representation of an integer
*
Motzkin number
In mathematics, the th Motzkin number is the number of different ways of drawing non-intersecting chords between points on a circle (not necessarily touching every point by a chord). The Motzkin numbers are named after Theodore Motzkin and have d ...
*
Multiplicities of entries in Pascal's triangle
*
Narayana number
In combinatorics, the Narayana numbers \operatorname(n, k), n \in \mathbb^+, 1 \le k \le n form a triangular array of natural numbers, called the Narayana triangle, that occur in various counting problems. They are named after Canadian mathemati ...
*
Star of David theorem
*
Sun's curious identity
*
Table of Newtonian series In mathematics, a Newtonian series, named after Isaac Newton, is a sum over a sequence a_n written in the form
:f(s) = \sum_^\infty (-1)^n a_n = \sum_^\infty \frac a_n
where
:
is the binomial coefficient and (s)_n is the falling factorial. N ...
*
Trinomial expansion
In mathematics, a trinomial expansion is the expansion of a power of a sum of three terms into monomials. The expansion is given by
:(a+b+c)^n = \sum_ \, a^i \, b^ \;\! c^k,
where is a nonnegative integer and the sum is taken over all combina ...
Notes
References
*
*
*
*
*
*
*
*
*
*
*
*
*
External links
*
*
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Combinatorics
Factorial and binomial topics
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Operations on numbers
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