In
mathematics, the binomial coefficients are the positive
integers that occur as
coefficient
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...
s 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 of the
binomial
Binomial may refer to:
In mathematics
*Binomial (polynomial), a polynomial with two terms
*Binomial coefficient, numbers appearing in the expansions of powers of binomials
*Binomial QMF, a perfect-reconstruction orthogonal wavelet decomposition
* ...
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 ...
; this coefficient can be computed by the multiplicative formula
:
which using
factorial
In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial:
\begin
n! &= n \times (n-1) \times (n-2) ...
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 ...
, satisfying the
recurrence relation
:
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
Andreas Freiherr von Ettingshausen (25 November 1796 – 25 May 1878) was an Austrian mathematician and physicist.
Biography
Ettingshausen studied philosophy and jurisprudence at the University of Vienna. In 1817, he joined the University of Vi ...
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 ...
). The earliest known detailed discussion of binomial coefficients is in a tenth-century commentary, by
Halayudha, on an ancient
Sanskrit text,
Pingala's ''Chandaḥśāstra''. The second earliest description of binomial coefficients is given by
Al-Karaji
( fa, ابو بکر محمد بن الحسن الکرجی; c. 953 – c. 1029) was a 10th-century Persian mathematician and engineer who flourished at Baghdad. He was born in Karaj, a city near Tehran. His three principal surviving works are ...
. 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 t ...
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 numbers (taken to include 0) ''n'' and ''k'', the binomial coefficient
can be defined as the
coefficient
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...
of the
monomial
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered:
# A monomial, also called power product, is a product of powers of variables with nonnegative integer expon ...
''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
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
),
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 t ...
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 represented ...
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, 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 ...
, 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.
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
In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial:
\begin
n! &= n \times (n-1) \times (n-2) ...
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 mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
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, 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 t ...
is the important
recurrence relation
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 ...
:
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 t ...
.
* 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 ...
.
* 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 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 ''t ...
s are
* The
binomial distribution
In probability theory and statistics, the binomial distribution with parameters ''n'' and ''p'' is the discrete probability distribution of the number of successes in a sequence of ''n'' independent experiments, each asking a yes–no questi ...
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 ''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 abili ...
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.
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 po ...
:
:
The
derivative 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 id ...
(that is, any field that contains the
rational numbers), 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 t ...
.) 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 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
In combinatorics, Vandermonde's identity (or Vandermonde's convolution) is the following identity for binomial coefficients:
:=\sum_^r
for any nonnegative integers ''r'', ''m'', ''n''. The identity is named after Alexandre-Théophile Vandermon ...
, 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
In mathematics the ''n''th central binomial coefficient is the particular binomial coefficient
: = \frac = \prod\limits_^\frac \textn \geq 0.
They are called central since they show up exactly in the middle of the even-numbered rows in Pascal ...
.
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
:
and its relative
:
Let ''F''(''n'') denote the ''n''-th
Fibonacci number.
Then
:
This can be proved by
induction 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
In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − × ÷), and functions (e.g., ''n''th roo ...
for
partial sums
:
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 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. It follows from
which is proved by
induction 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 numbers, indexed so that , then the identity
has the following combinatorial proof. One may show by
induction 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 t ...
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
In mathematics, Dixon's identity (or Dixon's theorem or Dixon's formula) is any of several different but closely related identities proved by A. C. Dixon, some involving finite sums of products of three binomial coefficients, and some evaluating ...
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 for a ...
to convert
trigonometric functions 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
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 serie ...
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 Kummer is a German surname. Notable people with the surname include:
* Bernhard Kummer (1897–1962), German Germanist
* Clare Kummer (1873—1958), American composer, lyricist and playwright
* Clarence Kummer (1899–1930), American jockey
* Chri ...
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
The fractional part or decimal part of a non‐negative real number x is the excess beyond that number's integer part. If the latter is defined as the largest integer not greater than , called floor of or \lfloor x\rfloor, its fractional part can ...
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 m ...
(1974) is that any integer divides
almost all
In mathematics, the term "almost all" means "all but a negligible amount". More precisely, if X is a set, "almost all elements of X" means "all elements of X but those in a negligible subset of X". The meaning of "negligible" depends on the math ...
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 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 limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Land ...
.
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, \do ...
and
is the
Euler–Mascheroni constant.)
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
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pr ...
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 po ...
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 divis ...
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 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 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 gre ...
, 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 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 ...
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 t ...
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 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
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
. 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. Th ...
,
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
typewriters and
computer terminals. Many
programming languages 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 mo ...
and the (related)
J programming language use the exclamation mark:
k ! n
. The binomial coefficient is implemented in
SciPy 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 (pr ...
:
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 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 t ...
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 A scheme is a systematic plan for the implementation of a certain idea.
Scheme or schemer may refer to:
Arts and entertainment
* ''The Scheme'' (TV series), a BBC Scotland documentary series
* The Scheme (band), an English pop band
* ''The Schem ...
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 t ...
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, implementatio ...
and Python's
SciPy 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 P ...
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. ...
. 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 coefficient ...
*
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
** ...
*
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 ...
*
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 Combinatorial enumeration, counting problems. They are named ...
*
Star of David theorem
*
Sun's curious identity
In combinatorics, Sun's curious identity is the following identity involving binomial coefficients, first established by Zhi-Wei Sun in 2002:
:
(x+m+1)\sum_^m(-1)^i\dbinom\dbinom
-\sum_^\dbinom(-4)^i=(x-m)\dbinom.
Proofs
After Sun's publicati ...
*
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. ...
*
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 combin ...
Notes
References
*
*
*
*
*
*
*
*
*
*
*
*
*
External links
*
*
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Combinatorics
Factorial and binomial topics
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