In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written
\tbinom{n}{k}.
\binomnk=
n x (n-1) x … x (n-k+1) | |
k x (k-1) x … x 1 |
,
\binom{n}{k}=
n! | |
k!(n-k)! |
.
For example, the fourth power of is
\begin{align} (1+x)4&=\tbinom{4}{0}x0+\tbinom{4}{1}x1+\tbinom{4}{2}x2+\tbinom{4}{3}x3+\tbinom{4}{4}x4\\ &=1+4x+6x2+4x3+x4, \end{align}
\tbinom{4}{2}=\tfrac{4 x 3}{2 x 1}=\tfrac{4!}{2!2!}=6
Arranging the numbers
\tbinom{n}{0},\tbinom{n}{1},\ldots,\tbinom{n}{n}
\binom{n}{k}=\binom{n-1}{k-1}+\binom{n-1}{k}.
The binomial coefficients occur in many areas of mathematics, and especially in combinatorics. The symbol
\tbinom{n}{k}
\tbinom{n}{k}
\tbinom{4}{2}=6
The binomial coefficients can be generalized to
\tbinom{z}{k}
Andreas von Ettingshausen introduced the notation
\tbinomnk
Alternative notations include,,,,, and, in all of which the stands for combinations 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.
0 | 1 | 2 | 3 | 4 | ⋯ | ||
---|---|---|---|---|---|---|---|
0 | 1 | ⋯ | |||||
1 | 1 | 1 | ⋯ | ||||
2 | 1 | 2 | 1 | ⋯ | |||
3 | 1 | 3 | 3 | 1 | ⋯ | ||
4 | 1 | 4 | 6 | 4 | 1 | ⋯ | |
⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋱ | |
The first few binomial coefficients on a left-aligned Pascal's triangle |
For natural numbers (taken to include 0) and, the binomial coefficient
\tbinomnk
Another occurrence of this number is in combinatorics, where it gives the number of ways, disregarding order, that objects can be chosen from among objects; more formally, the number of -element subsets (or -combinations) of an -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 factors of the power one temporarily labels the term with an index (running from to), then each subset of indices gives after expansion a contribution, and the coefficient of that monomial in the result will be the number of such subsets. This shows in particular that
\tbinomnk
\tbinomnk
k=a1+a2+ … +an
Several methods exist to compute the value of
\tbinom{n}{k}
One method uses the recursive, purely additive formula for all integers
n,k
1\lek<n,
The formula follows from considering the set and counting separately (a) the -element groupings that include a particular set element, say "", in every group (since "" 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 ""; this enumerates all the possible -combinations of elements. It also follows from tracing the contributions to Xk in . As there is zero or in, one might extend the definition beyond the above boundaries to include
\tbinomnk=0
A more efficient method to compute individual binomial coefficients is given by the formulawhere the numerator of the first fraction
n\underline{k
Due to the symmetry of the binomial coefficient with regard to and, calculation may be optimised by setting the upper limit of the product above to the smaller of and .
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 denotes the factorial of . 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 is small and 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,
See main article: Binomial series. The multiplicative formula allows the definition of binomial coefficients to be extended[2] 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
\tbinom\alphak
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,[3] 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.
See main article: Pascal's triangle and Pascal's rule.
Pascal's rule is the important recurrence relationwhich can be used to prove by mathematical induction that
\tbinomnk
Pascal's rule also gives rise to Pascal's triangle:
Row number contains the numbers
\tbinom{n}{k}
(x+y)5=\underline{1}x5+\underline{5}x4y+\underline{10}x3y2+\underline{10}x2y3+\underline{5}xy4+\underline{1}y5.
Binomial coefficients are of importance in combinatorics, because they provide ready formulas for certain frequent counting problems:
\tbinomnk
\tbinom{n+k-1}k
\tbinom{n+k}k
\tbinom{n+1}k
\tfrac{1}{n+1}\tbinom{2n}{n}.
\tbinomnkpk(1-p)n-k.
For any nonnegative integer k, the expression can be simplified and defined as a polynomial divided by :
\binom{t}{k}=
t\underline{k | |
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
\tbinom{t}{k}
Its coefficients are expressible in terms of Stirling numbers of the first kind:
\binom{t}{k}=
k | ||
\sum | s(k,i) | |
i=0 |
ti | |
k! |
.
\tbinom{t}{k}
d | |
dt |
\binom{t}{k}=\binom{t}{k}
k-1 | |
\sum | |
i=0 |
1 | |
t-i |
.
0
t-1
d | |
dt |
\binom{t}{k}=
k-1 | |
\sum | |
i=0 |
(-1)k-i-1 | |
k-i |
\binom{t}{i}.
Over any field of characteristic 0 (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 ak is the kth difference of the sequence p(0), p(1), ..., p(k). Explicitly,[5]
See main article: Integer-valued polynomial. Each polynomial
\tbinom{t}{k}
t
The integer-valued polynomial can be rewritten as
9\binom{t}{2}+6\binom{t}{1}+0\binom{t}{0}.
The factorial formula facilitates relating nearby binomial coefficients. For instance, if k is a positive integer and n is arbitrary, thenand, with a little more work,
\binom{n-1}{k}-\binom{n-1}{k-1}=
n-2k | |
n |
\binom{n}{k}.
We can also get
\binom{n-1}{k}=
n-k | |
n |
\binom{n}{k}.
Moreover, the following may be useful:
\binom{n}{h}\binom{n-h}{k}=\binom{n}{k}\binom{n-k}{h}=\binom{n}{h+k}\binom{h+k}{h}.
For constant n, we have the following recurrence:
\binom{n}{k}=
n-k+1 | |
k |
\binom{n}{k-1}.
To sum up, we have
\binom{n}{k}=\binomn{n-k}=
n-k+1 | |
k |
\binom{n}{k-1}=
n | |
n-k |
\binom{n-1}{k}=
n | |
k |
\binom{n-1}{k-1}=
n | |
n-2k |
(\binom{n-1}{k}-\binom{n-1}{k-1})=\binom{n-1}k+\binom{n-1}{k-1}.
The formulasays that 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 and . 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
2n
The formulasand
n | |
\sum | |
k=0 |
k2\binomnk=(n+n2)2n-2
The Chu–Vandermonde identity, which holds for any complex values m and n and any non-negative integer k, isand can be found by examination of the coefficient of
xk
Another form of the Chu–Vandermonde identity, which applies for any integers j, k, and n satisfying, isThe proof is similar, but uses the binomial series expansion with negative integer exponents.When, equation gives the hockey-stick identity
n | |
\sum | |
m=k |
\binommk=\binom{n+1}{k+1}
m | |
\sum | |
r=0 |
\binom{n+r}r=\binom{n+m+1}{m}.
Let F(n) denote the n-th Fibonacci number.Then
\lfloorn/2\rfloor | |
\sum | |
k=0 |
\binom{n-k}k=F(n+1).
For integers s and t such that
0\leqt<s,
\binom{n}{t}+\binom{n}{t+s}+\binom{n}{t+2s}+\ldots= | 1 |
s |
s-1 | ||
\sum | \left(2\cos | |
j=0 |
\pij | |
s |
| ||||
\right) |
.
For small, these series have particularly nice forms; for example,[6]
\binom{n}{0}+\binom{n}{3}+\binom{n}{6}+ … =
1 | |
3 |
\left(2n+2\cos
n\pi | |
3 |
\right)
\binom{n}{1}+\binom{n}{4}+\binom{n}{7}+ … =
1 | |
3 |
\left(2n+2\cos
(n-2)\pi | |
3 |
\right)
\binom{n}{2}+\binom{n}{5}+\binom{n}{8}+ … =
1 | |
3 |
\left(2n+2\cos
(n-4)\pi | |
3 |
\right)
\binom{n}{0}+\binom{n}{4}+\binom{n}{8}+ … =
1 | |
2 |
\left(2n-1
| |||||
+2 | \cos |
n\pi | |
4 |
\right)
\binom{n}{1}+\binom{n}{5}+\binom{n}{9}+ … =
1 | |
2 |
\left(2n-1
| |||||
+2 | \sin |
n\pi | |
4 |
\right)
\binom{n}{2}+\binom{n}{6}+\binom{n}{10}+ … =
1 | |
2 |
\left(2n-1
| |||||
-2 | \cos |
n\pi | |
4 |
\right)
\binom{n}{3}+\binom{n}{7}+\binom{n}{11}+ … =
1 | |
2 |
\left(2n-1
| |||||
-2 | \sin |
n\pi | |
4 |
\right)
Although there is no closed formula for partial sums
k | |
\sum | |
j=0 |
\binomnj
k | |
\sum | |
j=0 |
(-1)j\binom{n}{j}=(-1)k\binom{n-1}{k},
n | |
\sum | |
j=0 |
(-1)j\binomnj=0
n | |
\sum | |
j=0 |
(-1)j\binomnjP(j)=0.
P(x)=x(x-1) … (x-k+1)
When P(x) is of degree less than or equal to n,where
an
More generally for,
n | |
\sum | |
j=0 |
(-1)j\binomnjP(m+(n-j)d)=dnn!an
The series 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.
Many identities involving binomial coefficients can be proved by combinatorial means. For example, for nonnegative integers
{n}\geq{q}
n | |
\sum | |
k=q |
\binom{n}{k}\binom{k}{q}=2n-q\binom{n}{q}
\tbinomnq
2n-q
In Pascal's identity
{n\choosek}={n-1\choosek-1}+{n-1\choosek},
The identity also has a combinatorial proof. The identity reads
n | |
\sum | |
k=0 |
\binom{n}{k}2=\binom{2n}{n}.
Suppose you have
2n
\tbinom{2n}n
n-k
n\binom | |
\sum | |
k=0 |
nk\binomn{n-k}=\binom{2n}n.
If one denotes by the sequence of Fibonacci numbers, indexed so that, then the identityhas 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
\tbinom{n-k}{k}
The number of k-combinations for all k, , is the sum of the nth 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
2n-1
a | |
\sum | |
k=-a |
(-1)k{2a\choosek+a}3=
(3a)! | |
(a!)3 |
a(-1) | |
\sum | |
k=-a |
k{a+b\choosea+k}{b+c\chooseb+k}{c+a\choosec+k}=
(a+b+c)! | |
a!b!c! |
,
Certain trigonometric integrals have values expressible in terms of binomial coefficients: For any
m,n\in\N,
\pi | |
\int | |
-\pi |
\cos((2m-n)x)\cosn(x) dx=
\pi | |
2n-1 |
\binom{n}{m}
\pi | |
\int | |
-\pi |
\sin((2m-n)x)\sinn(x) dx= \begin{cases} (-1)m+(n+1)/2
\pi | |
2n-1 |
\binom{n}{m},&nodd\\ 0,&otherwise \end{cases}
\pi | |
\int | |
-\pi |
\cos((2m-n)x)\sinn(x) dx= \begin{cases} (-1)m+(n/2)
\pi | |
2n-1 |
\binom{n}{m},&neven\\ 0,&otherwise \end{cases}
These can be proved by using Euler's formula to convert trigonometric functions to complex exponentials, expanding using the binomial theorem, and integrating term by term.
If n is prime, then for every k with
0\leqk\leqn-1.
Indeed, we have
\binom{n-1}k={(n-1)(n-2) … (n-k)\over1 ⋅ 2 … k} =
k | |
\prod | |
i=1 |
{n-i\overi}\equiv
k | |
\prod | |
i=1 |
{-i\overi}=(-1)k\modn.
For a fixed, the ordinary generating function of the sequence
\tbinomn0,\tbinomn1,\tbinomn2,\ldots
infty | |
\sum | |
k=0 |
{n\choosek}xk=(1+x)n.
For a fixed, the ordinary generating function of the sequence
\tbinom0k,\tbinom1k,\tbinom2k,\ldots,
infty | |
\sum | |
n=0 |
{n\choosek}yn=
yk | |
(1-y)k+1 |
.
The bivariate generating function of the binomial coefficients is
infty | |
\sum | |
n=0 |
n | |
\sum | |
k=0 |
{n\choosek}xkyn=
1 | |
1-y-xy |
.
A symmetric bivariate generating function of the binomial coefficients is
infty | |
\sum | |
n=0 |
infty | |
\sum | |
k=0 |
{n+k\choosek}xkyn=
1 | |
1-x-y |
.
x\toxy
A symmetric exponential bivariate generating function of the binomial coefficients is:
infty | |
\sum | |
n=0 |
infty | |
\sum | |
k=0 |
{n+k\choosek}
xkyn | |
(n+k)! |
=ex+y.
See main article: Kummer's theorem and Lucas' theorem. 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
\tbinom{m+n}{m}
\tbinomnk
\tbinomnk
\tbinom{pr}{s}
\tbinom{9}{6}
A somewhat surprising result by David Singmaster (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
\tbinomnk
\tbinomnk
\limN\toinfty
f(N) | |
N(N+1)/2 |
=1.
\tbinomnk
Binomial coefficients have divisibility properties related to least common multiples of consecutive integers. For example:[10]
\binom{n+k}k
\operatorname{lcm | |
(n,n+1,\ldots,n+k)}n |
\binom{n+k}k
\operatorname{lcm | |
(n,n+1,\ldots,n+k)}{n ⋅ |
\operatorname{lcm}(\binom{k}0,\binom{k}1,\ldots,\binom{k}k)}
Another fact:An integer is prime if and only ifall the intermediate binomial coefficients
\binomn1,\binomn2,\ldots,\binomn{n-1}
Proof:When p is prime, p divides
\binompk=
p ⋅ (p-1) … (p-k+1) | |
k ⋅ (k-1) … 1 |
\tbinompk
\binomnp=
n(n-1)(n-2) … (n-p+1) | = | |
p! |
k(n-1)(n-2) … (n-p+1) | |
(p-1)! |
\not\equiv0\pmod{n}
The following bounds for
\tbinomnk
k
From the divisibility properties we can infer thatwhere both equalities can be achieved.
The following bounds are useful in information theory:[11] where
H(p)=-plog2(p)-(1-p)log2(1-p)
1\leqk\leqn-1
Stirling's approximation yields the following approximation, valid when
n-k,k
n
\sqrt{n}{2n\choosen}\ge22n-1
If n is large and k is linear in n, various precise asymptotic estimates exist for the binomial coefficient . For example, if
|n/2-k|=o(n2/3)
If is large and is (that is, if), thenwhere again is the little o notation.[14]
A simple and rough upper bound for the sum of binomial coefficients can be obtained using the binomial theorem:More precise bounds are given byvalid for all integers
n>k\geq1
\varepsilon
eq |
k/n\leq1/2
The infinite product formula for the gamma function also gives an expression for binomial coefficientswhich yields the asymptotic formulasas
k\toinfty
This asymptotic behaviour is contained in the approximationas well. (Here
Hk
\gamma
Further, the asymptotic formulahold true, whenever
k\toinfty
j/k\tox
x
See main article: Multinomial theorem. Binomial coefficients can be generalized to multinomial coefficients defined to be the number:
{n\choosek1,k2,\ldots,kr}=
n! | |
k1!k2! … kr! |
rk | |
\sum | |
i=n. |
While the binomial coefficients represent the coefficients of, the multinomial coefficientsrepresent the coefficients of the polynomial
(x1+x2+ … +
n. | |
x | |
r) |
{n\choosek1,k2}={n\choosek1,n-k1}={n\choosek1}={n\choosek2}.
The combinatorial interpretation of multinomial coefficients is distribution of n distinguishable elements over r (distinguishable) containers, each containing exactly ki 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:
{n\choosek1,k2,\ldots,kr}={n-1\choosek1-1,k2,\ldots,kr}+{n-1\choosek1,k2-1,\ldots,kr}+\ldots+{n-1\choosek1,k2,\ldots,kr-1}
{n\choosek1,k2,\ldots,kr}={n\choose
k | |
\sigma1 |
,k | |
\sigma2 |
,\ldots,k | |
\sigmar |
(\sigmai)
Using Stirling numbers of the first kind the series expansion around any arbitrarily chosen point
z0
\begin{align}{z\choosek}=
1 | |
k! |
k | |
\sum | |
i=0 |
zisk,i
k | |
&=\sum | |
i=0 |
(z-
i | |
z | |
0) |
k | |
\sum | |
j=i |
{z0\choosej-i}
sk+i-j,i | |
(k+i-j)! |
k | |
\ &=\sum | |
i=0 |
i | |
(z-z | |
0) |
k | |
\sum | |
j=i |
j-i | |
z | |
0 |
{j\choosei}
sk,j | |
k! |
.\end{align}
The definition of the binomial coefficients can be extended to the case where
n
k
In particular, the following identity holds for any non-negative integer
k
{{1/2}\choose{k}}={{2k}\choose{k}} | (-1)k+1 |
22k(2k-1) |
.
This shows up when expanding
\sqrt{1+x}
\sqrt{1+x}=\sumk\geq{\binom{1/2}{k}}xk.
One can express the product of two binomial coefficients as a linear combination of binomial coefficients:
{z\choosem}{z\choosen}=
min(m,n) | |
\sum | |
k=0 |
{m+n-k\choosek,m-k,n-k}{z\choosem+n-k},
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 nth row of the Pascal triangle is given by the formula:
n | |
\prod | |
k=0 |
n | |
\binom{n}{k}=\prod | |
k=1 |
k2k-n-1.
The partial fraction decomposition of the reciprocal is given by
1 | |
{z\choosen |
See main article: Binomial series. Newton's binomial series, named after Sir Isaac Newton, is a generalization of the binomial theorem to infinite series:
(1+z)\alpha=
infty | |
\sum | |
n=0 |
{\alpha\choosen}zn=1+{\alpha\choose1}z+{\alpha\choose2}z2+ … .
The identity can be obtained by showing that both sides satisfy the differential equation .
The radius of convergence of this series is 1. An alternative expression is
1 | |
(1-z)\alpha+1 |
=
infty | |
\sum | |
n=0 |
{n+\alpha\choosen}zn
{n\choosek}=(-1)k{k-n-1\choosek}
Binomial coefficients count subsets of prescribed size from a given set. A related combinatorial problem is to count multisets 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;[16] 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 ruleOne possible alternative characterization of this identity is as follows:We may define the falling factorial asand the corresponding rising factorial asso, for example,Then the binomial coefficients may be written aswhile the corresponding multiset coefficient is defined by replacing the falling with the rising factorial:
For any n,
\begin{align}\binom{-n}{k}&=
-n ⋅ -(n+1)...-(n+k-2) ⋅ -(n+k-1) | |
k! |
| ||||
\\ &=(-1) |
\\ &=(-1)k\binom{n+k-1}{k}\\ &=(-1)k\left(\binom{n}{k}\right) .\end{align}
n=-1
(-1)k=\binom{-1}{k}=\left(\binom{-k}{k}\right).
For example, if n = −4 and k = 7, then r = 4 and f = 10:
\begin{align}\binom{-4}{7}&=
-10 ⋅ -9 ⋅ -8 ⋅ -7 ⋅ -6 ⋅ -5 ⋅ -4 | |
1 ⋅ 2 ⋅ 3 ⋅ 4 ⋅ 5 ⋅ 6 ⋅ 7 |
| ||||
\\ &=(-1) |
\\ &=\left(\binom{-7}{7}\right)\left(\binom{4}{7}\right)=\binom{-1}{7}\binom{10}{7}.\end{align}
The binomial coefficient is generalized to two real or complex valued arguments using the gamma function or beta function via
{x\choosey}=
\Gamma(x+1) | |
\Gamma(y+1)\Gamma(x-y+1) |
=
1 | |
(x+1)\Beta(y+1,x-y+1) |
.
\Gamma
{x\choosey}=
\sin(y\pi) | |
\sin(x\pi) |
{-y-1\choose-x-1}=
\sin((x-y)\pi) | |
\sin(x\pi) |
{y-x-1\choosey};
{x\choosey} ⋅ {y\choosex}=
\sin((x-y)\pi) | |
(x-y)\pi |
.
The resulting function has been little-studied, apparently first being graphed in . Notably, many binomial identities fail: but for n positive (so
-n
y=x
0\leqy\leqx
0\leqx\leqy
x\geq0,y\leq0
x\leq0,y\geq0
0>x>y
-1>y>x+1
The binomial coefficient has a q-analog generalization known as the Gaussian binomial coefficient.
The definition of the binomial coefficient can be generalized to infinite cardinals by defining:
{\alpha\choose\beta}=\left|\left\{B\subseteqA:\left|B\right|=\beta\right\}\right|
\alpha
\alpha
Assuming the Axiom of Choice, one can show that for any infinite cardinal
\alpha
\tbinomnk=0
k<0
\tbinomnk
k<0
\alpha=n
style\binom{n}{k}=0
k>n
(k=n+1)
n-(n+1)+1=0
k
k\geqn+1