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Number of subsets of a given size
mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed
Binomial_coefficient
Algebraic expansion of powers of a binomial
} The coefficient a {\displaystyle a} in each term a x k y m {\displaystyle \textstyle ax^{k}y^{m}} is known as the binomial coefficient ( n
Binomial_theorem
Sequence of numbers ((2n) choose (n))
In mathematics the nth central binomial coefficient is the particular binomial coefficient ( 2 n n ) = ( 2 n ) ! ( n ! ) 2 for all n ≥ 0. {\displaystyle
Central_binomial_coefficient
Family of polynomials
the Gaussian binomial coefficients (also called Gaussian coefficients, Gaussian numbers, Gaussian polynomials, or q-binomial coefficients) are q-analogs
Gaussian_binomial_coefficient
Probability distribution
positive covariance term. The term "negative binomial" is likely due to the fact that a certain binomial coefficient that appears in the formula for the probability
Negative binomial distribution
Negative_binomial_distribution
Mathematical series
the right-hand side is expressed in terms of the (generalized) binomial coefficients ( α k ) = α ( α − 1 ) ( α − 2 ) ⋯ ( α − k + 1 ) k ! . {\displaystyle
Binomial_series
Triangular array of the binomial coefficients
mathematics, Pascal's triangle is an infinite triangular array of the binomial coefficients which play a crucial role in probability theory, combinatorics,
Pascal's_triangle
Multiplicative factor in a mathematical expression
v=x_{1}e_{1}+x_{2}e_{2}+\dotsb +x_{n}e_{n}.} Correlation coefficient Degree of a polynomial Monic polynomial Binomial coefficient "ISO 80000-1:2009". International Organization
Coefficient
Probability distribution
! {\displaystyle {\binom {n}{k}}={\frac {n!}{k!(n-k)!}}} is the binomial coefficient. The formula can be understood as follows: pk qn−k is the probability
Binomial_distribution
Number theory theorem
theory, Lucas's theorem expresses the remainder of division of the binomial coefficient ( m n ) {\displaystyle {\tbinom {m}{n}}} by a prime number p in terms
Lucas's_theorem
Selection of items from a set
{\displaystyle C(n,k)} or C k n {\displaystyle C_{k}^{n}} , is equal to the binomial coefficient: ( n k ) = n ( n − 1 ) ⋯ ( n − k + 1 ) k ( k − 1 ) ⋯ 1 , {\displaystyle
Combination
Theorem on powers of primes dividing binomial coefficients
number p that divides a given binomial coefficient. In other words, it gives the p-adic valuation of a binomial coefficient. The theorem is named after
Kummer's_theorem
Average uncertainty in variable's states
In information theory, the entropy of a random variable quantifies the average level of uncertainty or information associated with the variable's potential
Entropy_(information_theory)
Addition of several numbers or other values
Bernoulli number, and ( p k ) {\displaystyle {\binom {p}{k}}} is a binomial coefficient. In the following summations, a is assumed to be different from 1
Summation
Arrangement of trinomial coefficients
contains the binomial coefficients that appear in the binomial expansion and the binomial distribution. The binomial and trinomial coefficients, expansions
Pascal's_pyramid
Conjecture in combinatorial number theory
times, as do all central binomial coefficients except for 1 and 2; (it is in principle not excluded that such a coefficient would appear five, seven,
Singmaster's_conjecture
Graphical aid for deriving some concepts in combinatorics
distinguishable bins. The solution to this particular problem is given by the binomial coefficient ( n + k − 1 k − 1 ) {\displaystyle {\tbinom {n+k-1}{k-1}}} , which
Stars and bars (combinatorics)
Stars_and_bars_(combinatorics)
Pattern defining an infinite sequence of numbers
example of a multidimensional recurrence relation is given by the binomial coefficients ( n k ) {\displaystyle {\tbinom {n}{k}}} , which count the ways
Recurrence_relation
Number of partitions of an integer
( N , M , n ) {\displaystyle p(N,M,n)} is the following Gaussian binomial coefficient: ∑ n = 0 ∞ p ( N , M , n ) q n = ( N + M M ) q = ( 1 − q N + M )
Partition function (number theory)
Partition_function_(number_theory)
Formal power series
function for binomial coefficients for a fixed n, one may ask for a bivariate generating function that generates the binomial coefficients (n k) for all
Generating_function
Mathematical set with repetitions allowed
Like the binomial distribution that involves binomial coefficients, there is a negative binomial distribution in which the multiset coefficients occur.
Multiset
In mathematics, a polynomial with two terms
is an ideal that is generated by binomials that are difference of monomials; that is, binomials whose two coefficients are 1 and −1. A toric variety is
Binomial_(polynomial)
Topics referred to by the same term
Look up binomial in Wiktionary, the free dictionary. Binomial may refer to: Binomial (polynomial), a polynomial with two terms Binomial coefficient, numbers
Binomial
Decomposition of an integer as a sum of positive integers
partition yields a partition of n − M into at most M parts. The Gaussian binomial coefficient is defined as: ( k + ℓ ℓ ) q = ( k + ℓ k ) q = ∏ j = 1 k + ℓ ( 1
Integer_partition
Combinatorial identity about binomial coefficients
Pascal's formula) is a combinatorial identity about binomial coefficients. The binomial coefficients are the numbers that appear in Pascal's triangle. Pascal's
Pascal's_rule
Transformation of a mathematical sequence
In combinatorics, the binomial transform is a sequence transformation (i.e., a transform of a sequence) that computes its forward differences. It is closely
Binomial_transform
Bhargava factorial Binomial coefficient Pascal's triangle Binomial distribution Binomial proportion confidence interval Binomial-QMF (Daubechies wavelet
List of factorial and binomial topics
List_of_factorial_and_binomial_topics
Mathematical fallacy
endomorphism. One way to prove this is to show that p divides all the binomial coefficients except for the first and the last, so all the intermediate terms
Freshman's_dream
Generalization of the binomial theorem to other polynomials
theorem are the multinomial coefficients. They can be expressed in numerous ways, including as a product of binomial coefficients or of factorials: ( n k
Multinomial_theorem
Natural number
composite number, an Erdős–Woods number, a Pell number, a central binomial coefficient, and a primitive abundant number. 70 is the smallest weird number
70_(number)
Data structure that acts as a priority queue
binomial tree of order k {\displaystyle k} has ( k d ) {\displaystyle {\tbinom {k}{d}}} nodes at depth d {\displaystyle d} , a binomial coefficient.
Binomial_heap
Branch of discrete mathematics
astronomer Rabbi Abraham ibn Ezra (c. 1140) established the symmetry of binomial coefficients, while a closed formula was obtained later by the talmudist and
Combinatorics
_{s}(z)} is a polylogarithm. ( n k ) {\displaystyle n \choose k} is binomial coefficient exp ( x ) {\displaystyle \exp(x)} denotes exponential of x {\displaystyle
List_of_mathematical_series
Extension of the factorial function
of binomial coefficients motivates why the properties of the gamma function when extended to negative numbers are natural. A binomial coefficient gives
Gamma_function
Any experiment with two possible random outcomes
) {\displaystyle {n \choose k}} is a binomial coefficient. Bernoulli trials may also lead to negative binomial distributions (which count the number
Bernoulli_trial
Mathematical function
special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral B ( z 1 , z 2 ) = ∫ 0 1 t z 1 − 1
Beta_function
Product of numbers from 1 to n
sequences are closely related to the factorials, including the binomial coefficients, double factorials, falling factorials, primorials, and subfactorials
Factorial
Mathematical identity involving sums of binomial coefficients
Abel's binomial theorem, named after Niels Henrik Abel, is a mathematical identity involving sums of binomial coefficients. It states the following: ∑
Abel's_binomial_theorem
Series related to Ramanujan's pi formulas
recurrence relation, sequences which may be expressed in terms of binomial coefficients ( n k ) {\displaystyle {\tbinom {n}{k}}} , and A , B , C {\displaystyle
Ramanujan–Sato_series
Technique for proving sets have equal size
powerful insights into each or both of the sets. The symmetry of the binomial coefficients states that ( n k ) = ( n n − k ) . {\displaystyle {n \choose k}={n
Bijective_proof
Statistic for rank correlation
n − 1 ) 2 {\displaystyle {n \choose 2}={n(n-1) \over 2}} is the binomial coefficient for the number of ways to choose two items from n items. The number
Kendall rank correlation coefficient
Kendall_rank_correlation_coefficient
American mathematician (1927–2010)
Selfridge, J. L. (1993). "Estimates of the least prime factor of a binomial coefficient". Mathematics of Computation. 61 (203): 215–224. Bibcode:1993MaCom
John_Selfridge
the binomial coefficient with n = 1 / 2 {\displaystyle n=1/2} , but it may also be written in terms of the double factorial or integer binomial coefficients:
Perimeter_of_an_ellipse
Special type of prime number
1{\pmod {p^{4}}},} where the expression in left-hand side denotes a binomial coefficient. In comparison, Wolstenholme's theorem states that for every prime
Wolstenholme_prime
Statistical rule of thumb
sample should result in a histogram with bin counts given by the binomial coefficients. Since the total sample size is fixed to n {\displaystyle n} we
Sturges's_rule
Rational number sequence
{B_{k}^{+}n^{m+1-k}}{k!(m+1-k)!}},} where (m + 1 k) denotes the binomial coefficient. For example, taking m to be 1 gives the triangular numbers 0, 1
Bernoulli_number
Solved prime-number problem
{\displaystyle p^{r}} in the prime decomposition of the central binomial coefficient ( 2 n n ) = ( 2 n ) ! / ( n ! ) 2 {\displaystyle \textstyle {\binom
Proof_of_Bertrand's_postulate
French polymath (1623–1662)
Kieren MacMillan, Jonathan Sondow (2011). "Proofs of power sum and binomial coefficient congruences via Pascal's identity". American Mathematical Monthly
Blaise_Pascal
Natural number
and preceding 127. As the binomial coefficient ( 9 4 ) {\displaystyle {\tbinom {9}{4}}} , 126 is a central binomial coefficient, and in Pascal's Triangle
126_(number)
Natural number
OEIS Foundation. Retrieved 2016-05-31. "Sloane's A000332 : Binomial coefficient binomial(n,4) = n*(n-1)*(n-2)*(n-3)/24". The On-Line Encyclopedia of
35_(number)
In mathematics, a binomial ring is a commutative ring whose additive group is torsion-free and contains all binomial coefficients ( x n ) = x ( x − 1
Binomial_ring
Mathematical functions
{\displaystyle (x)_{n}} with yet another meaning, namely to denote the binomial coefficient ( x n ) {\displaystyle {\tbinom {x}{n}}} . In this article, the symbol
Falling_and_rising_factorials
Expression for sums of powers
} Here, ( p + 1 r ) {\textstyle {\binom {p+1}{r}}} is the binomial coefficient " p + 1 {\displaystyle p+1} choose r {\displaystyle r} ", and the
Faulhaber's_formula
Device invented by Francis Galton
the number of paths to the kth bin on the bottom is given by the binomial coefficient ( n k ) {\displaystyle {n \choose k}} . Note that the leftmost bin
Galton_board
published in 2016. The Erdős squarefree conjecture that central binomial coefficients C(2n, n) are never squarefree for n > 4 was proved in 1996 by Olivier
List of conjectures by Paul Erdős
List_of_conjectures_by_Paul_Erdős
(Daubechies wavelet). It was an extension of Akansu's prior work on Binomial coefficient and Hermite polynomials wherein he developed the Modified Hermite
Binomial_QMF
Recursive integer sequence
Catalan number can be expressed directly in terms of the central binomial coefficients by C n = 1 n + 1 ( 2 n n ) = ( 2 n ) ! ( n + 1 ) ! n ! for n ≥
Catalan_number
Mathematical result on arithmetic properties of binomial coefficients
properties of binomial coefficients. It was discovered by Henry W. Gould in 1972. The greatest common divisors of the binomial coefficients forming each
Star_of_David_theorem
Generalization of the product rule in calculus
− k ) ! {\displaystyle {n \choose k}={n! \over k!(n-k)!}} is the binomial coefficient and f ( j ) {\displaystyle f^{(j)}} denotes the j-th derivative of
General_Leibniz_rule
brackets. ( ◻ ◻ ) {\displaystyle {\binom {\Box }{\Box }}} Denotes a binomial coefficient: Given two nonnegative integers, ( n k ) {\displaystyle {\binom {n}{k}}}
Glossary of mathematical symbols
Glossary_of_mathematical_symbols
Probability of shared birthdays
where ! is the factorial operator, (365 n) is the binomial coefficient and kPr denotes permutation. The equation expresses the fact that
Birthday_problem
(for α, β, m, n integers with β ≠ 0 and m, n ≥ 0, see also Binomial coefficient) ∫ − t t sin m ( α x ) cos n ( β x ) d x = 0 {\displaystyle \int
Lists_of_integrals
(1971). "Research Problems: How often does an integer occur as a binomial coefficient?". American Mathematical Monthly. 78 (4): 385–386. doi:10.2307/2316907
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Polynomial with integer value for integer input
the binomial coefficients. In other words, every integer-valued polynomial can be written as an integer linear combination of binomial coefficients in
Integer-valued_polynomial
Algebra associated to any vector space
V ) {\displaystyle \textstyle \bigwedge ^{\!k}(V)} is equal to a binomial coefficient: dim ⋀ k ( V ) = ( n k ) , {\displaystyle \dim {\textstyle \bigwedge
Exterior_algebra
Inequality in mathematics
order. The denominator is the number of terms in the numerator, the binomial coefficient ( n k ) . {\displaystyle {\tbinom {n}{k}}.} Maclaurin's inequality
Maclaurin's_inequality
Austrian mathematician and physicist (1796–1878)
{\binom {n}{k}}} for the binomial coefficient, which is the coefficient of x k {\displaystyle x^{k}} in the expansion of the binomial ( 1 + x ) k {\displaystyle
Andreas_von_Ettingshausen
Integer sequence
from n = 0) gives the highest power of 2 that divides the central binomial coefficient ( 2 n n ) {\displaystyle {\tbinom {2n}{n}}} , and it gives the numerator
Gould's_sequence
Sequence of end-to-end vectors across points of a lattice
connections to the number of combinations, which are counted by the binomial coefficient, and arranged in Pascal's triangle. The diagram below demonstrates
Lattice_path
Identity in mathematical combinatorics
of the Chu–Vandermonde identity. Using standard notation for q-binomial coefficients, the identity states that ( m + n k ) q = ∑ j ( m k − j ) q ( n
Q-Vandermonde_identity
binomial coefficient when the exponent is a prime p: ( p i ) = p ! i ! ( p − i ) ! {\displaystyle {p \choose i}={\frac {p!}{i!(p-i)!}}} The binomial coefficients
Proofs of Fermat's little theorem
Proofs_of_Fermat's_little_theorem
Result in number theory
p-1}\equiv 1{\pmod {p^{3}}}} holds, where the parentheses denote a binomial coefficient. For example, with p = 7, this says that 1716 is one more than a
Wolstenholme's_theorem
Measure of linear correlation
statistics, the Pearson correlation coefficient (PCC), also known as Pearson's r, the Pearson product-moment correlation coefficient (PPMCC), or simply the unqualified
Pearson correlation coefficient
Pearson_correlation_coefficient
Natural number
also the 16th area of a crystagon, equivalent with the quotient of binomial coefficient C ( 7 n , 2 ) {\displaystyle \mathrm {C} (7n,2)} and 7 {\displaystyle
888_(number)
Curve used in computer graphics and related fields
Bernstein basis polynomials of degree n. t0 = 1, (1 − t)0 = 1, and the binomial coefficient, ( n i ) {\displaystyle \scriptstyle {n \choose i}} , is: ( n i )
Bézier_curve
Natural number
OEIS Foundation. Retrieved 11 June 2016. "Sloane's A000332 : Binomial coefficient binomial(n,4)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation
700_(number)
Mathematical series, portmanteau of "Fibonacci" and "factorial"
Fibonomial coefficients (or Fibonacci-binomial coefficients) similarly as the factorial numbers are used in the definition of binomial coefficients. The series
Fibonorial
Transcendental single-variable function
hypergeometric series, summations involving the inverse of the central binomial coefficient, sums of the polygamma function, and Dirichlet L-series. The Clausen
Clausen_function
Systematic classification of 12 related enumerative problems concerning two finite sets
ways to partition a set of n elements into k non-empty subsets the binomial coefficient ( n k ) = n k _ k ! {\textstyle {\binom {n}{k}}={\frac {n^{\underline
Twelvefold_way
States of matter for water as a solid
{4}{2}}=6} allowed configurations of hydrogens for this oxygen atom (see Binomial coefficient). Thus, there are 6N/2 configurations that satisfy these N/2 atoms
Phases_of_ice
Fractal composed of triangles
zero with increasing n, a corollary is that the proportion of odd binomial coefficients tends to zero as n tends to infinity. The Towers of Hanoi puzzle
Sierpiński_triangle
Description of the behaviour of bosons
process, we can see that w ( n , g ) {\displaystyle w(n,g)} is just a binomial coefficient (See Notes below) w ( n , g ) = ( n + g − 1 ) ! n ! ( g − 1 ) !
Bose–Einstein_statistics
Sufficient condition for polynomial irreducibility
all of whose non-leading coefficients are divisible by p by properties of binomial coefficients, and whose constant coefficient is equal to p, and therefore
Eisenstein's_criterion
Type of mathematical generalization
to define the q-binomial coefficients, also known as Gaussian coefficients, Gaussian polynomials, or Gaussian binomial coefficients: ( n k ) q = [ n
Q-analog
Multiplicative function in number theory
{\displaystyle \mu _{k}\left(p^{a}\right)=(-1)^{a}{\binom {k}{a}}} where the binomial coefficient is taken to be zero if a > k {\displaystyle a>k} . The definition
Möbius_function
Type of analog or digital filter
polynomial arithmetic and uses binomial coefficients. The algorithm is extremely efficient if the Binomial coefficients are implemented from a look-up
Chebyshev_filter
Empirical law on the variance of species in a habitat
the Sundt-Jewel family are the Poisson, binomial, negative binomial (Pascal), extended truncated negative binomial and logarithmic series distributions.
Taylor's_law
Mathematical theorem on convolved binomial coefficients
identity (or Vandermonde's convolution) is the following identity for binomial coefficients: ( m + n r ) = ∑ k = 0 r ( m k ) ( n r − k ) {\displaystyle {m+n
Vandermonde's_identity
Number without repeated prime factors
sequences A019565, A048672 and A064273 in the OEIS.) The central binomial coefficient ( 2 n n ) {\displaystyle {2n \choose n}} is never squarefree for
Square-free_integer
Rational numbers in a reciprocal logarithm
integral logarithm and ( k m ) {\displaystyle {\tbinom {k}{m}}} is the binomial coefficient. It is also known that the zeta function, the gamma function, the
Gregory_coefficients
Mathematical approximation of a function
_{n=0}^{\infty }{\binom {\alpha }{n}}x^{n}} whose coefficients are the generalized binomial coefficients ( α n ) = ∏ k = 1 n α − k + 1 k = α ( α − 1 ) ⋯
Taylor_series
these are the only known Wilson primes. Primes p for which the binomial coefficient ( 2 p − 1 p − 1 ) ≡ 1 ( mod p 4 ) . {\displaystyle {{2p-1} \choose
List_of_prime_numbers
Theoretical limit on rate of mutation
sequence: Note that the number of sequences for distance d is just the binomial coefficient ( L d ) {\displaystyle {\tbinom {L}{d}}} for L=3, and that each sequence
Error_threshold_(evolution)
Type of proof technique
binomial coefficient ( n k ) . {\displaystyle {n \choose k}.} Therefore the total number of possible committees is the sum of binomial coefficients over
Double counting (proof technique)
Double_counting_(proof_technique)
sums and alternating sums of binomial coefficients. It has been suggested that he may have also discovered the binomial theorem in this context. 3rd century
Timeline of scientific discoveries
Timeline_of_scientific_discoveries
Online database of integer sequences
primes, the Fibonacci sequence, the lazy caterer's sequence, and the coefficients in the series expansion of ζ ( n + 2 ) ζ ( n ) {\displaystyle \textstyle
On-Line Encyclopedia of Integer Sequences
On-Line_Encyclopedia_of_Integer_Sequences
Generalization in fractional calculus
\left(a+1\right)}{\Gamma \left(b+1\right)\cdot \Gamma \left(a-b+1\right)}}} is the binomial coefficient. Caputo-type fractional derivative is closely related to the Riemann–Liouville
Caputo_fractional_derivative
Mathematical set of all subsets of a set
so the number of combinations, denoted as C(n, k) (also called binomial coefficient) is a number of subsets with k elements in a set with n elements;
Power_set
Orthogonal wavelets
processing perspective. It was an extension of the prior work on binomial coefficient and Hermite polynomials that led to the development of the Modified
Daubechies_wavelet
Indian inventions
Central Europe. London: W. H. Allen & co. Fowler, David (1996). "Binomial Coefficient Function". The American Mathematical Monthly. 103 (1): 1–17. doi:10
List of Indian inventions and discoveries
List_of_Indian_inventions_and_discoveries
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