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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
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
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
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
Probability distribution
since the binomial coefficient (nk) counts the number of ways to choose the positions of the k successes among the n trials. The binomial distribution
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
Probability distribution
of the probability mass function's binomial coefficient arises when considering the equivalent multiset coefficient ( ( r k ) ) {\displaystyle \textstyle
Negative binomial distribution
Negative_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
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−1k−1), which is the number of subsets of size k − 1 that can
Stars and bars (combinatorics)
Stars_and_bars_(combinatorics)
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
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
Transformation of a mathematical sequence
d ) ) {\displaystyle {\binom {\nu _{p}(n)}{\nu _{p}(d)}}} is the binomial coefficient. This convolution appears in the book by P. J. McCarthy (1986) and
Binomial_transform
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
Type of polynomial sequence
binomial type. Eric Temple Bell called these the "exponential polynomials" and that term is also sometimes seen in the literature. The coefficients S(n
Binomial_type
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)
Addition of several numbers or other values
where Bk denotes a Bernoulli number, and (pk) is a binomial coefficient. In the following summations, a is assumed to be different from 1
Summation
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
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
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
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
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
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)
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
_{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
(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
Mathematical fallacy
are less than p and p is prime. Since a binomial coefficient is always an integer, the nth binomial coefficient is divisible by p and hence equal to 0
Freshman's_dream
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
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
Any experiment with two possible random outcomes
scheme Bernoulli sampling Bernoulli distribution Binomial distribution Binomial coefficient Binomial proportion confidence interval Poisson sampling Sampling
Bernoulli_trial
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)
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
Solved prime-number problem
prime power factor pr in the prime decomposition of the central binomial coefficient (2nn)=(2n)!/(n!)2 is at most 2n; then, one shows that every prime
Proof_of_Bertrand's_postulate
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
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)
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
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
Natural number
composite number, an Erdős–Woods number, a Pell number, a central binomial coefficient, a primitive abundant number, and the 7th pentagonal number. 70 is
70_(number)
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
Mathematical theorem on convolved binomial coefficients
identity (or Vandermonde's convolution) is the following identity for binomial coefficients: (m+nr)=∑k=0r(mk)(nr−k) for any nonnegative integers r, m, n. The
Vandermonde's_identity
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
Recurrence relations of binomial coefficients in Pascal's triangle
Similarly, we find that the coefficient of x r {\displaystyle x^{r}} on the right hand side is given by the coefficient of x r + 1 {\displaystyle x^{r+1}}
Hockey-stick_identity
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
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
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
Type of permutation of a set of elements
and binomial topics Combinatorial principles Core Addition principle · Rule of product · Binomial coefficient · Pascal's triangle · Central binomial coefficient
Derangement
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
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
Expression for sums of powers of integers
formula is ∑k=1nkp=1p+1∑r=0p(p+1r)Br+np+1−r. Here, (p+1r) is the binomial coefficient "p+1 choose r", and the Br+ are the second Bernoulli numbers, identical
Faulhaber's_formula
Formula in mathematics
application of the binomial expansion, introducing another summation over the index s {\displaystyle s} . The product of the two binomial coefficients is simplified
Trinomial_expansion
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
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
Mathematical function
{s-1}{k}}\cdots ,\quad \Re (s)>0.\end{aligned}}} where (s k) is the binomial coefficient. It may also be generalized to ψ ( s + 1 ) = − γ − 1 m ∑ k = 1 m
Digamma_function
Recursive integer sequence
-th Catalan number can be expressed directly in terms of central binomial coefficients by C n = 1 n + 1 ( 2 n n ) = ( 2 n ) ! ( n + 1 ) ! n ! for n ≥
Catalan_number
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
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
(for α, β, m, n integers with β ≠ 0 and m, n ≥ 0, see also Binomial coefficient) ∫−ttsinm(αx)cosn(βx)dx=0 (for α, β real, n a non-negative integer
Lists_of_integrals
Numerical method for calculating the gamma function
arbitrarily subject to the restriction that Re(z + g + 1/2) > 0. The coefficients p, which depend on g, are slightly more difficult to calculate (see below)
Lanczos_approximation
{\displaystyle 2kn+1} if it is odd. Some people write "binomial number" when they mean binomial coefficient, but this usage is not standard and is deprecated
Binomial_number
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
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
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
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
Natural number
0 786 might be the largest n for which the value of the central binomial coefficient 2nCn is not divisible by an odd prime squared. If there is a larger
786_(number)
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
Upper bound on intersecting set families
number of sets in A {\displaystyle {\mathcal {A}}} is at most the binomial coefficient ( n − 1 r − 1 ) . {\displaystyle {\binom {n-1}{r-1}}.} The requirement
Erdős–Ko–Rado_theorem
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
Device invented by Francis Galton
the number of paths to the kth bin on the bottom is given by the binomial coefficient (nk). Note that the leftmost bin is the 0-bin, next to it is the
Galton_board
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
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)
Generalization of the binomial distribution
needed] Similarly, just like one can interpret the binomial distribution as the polynomial coefficients of ( p + q ) n {\displaystyle (p+q)^{n}} when expanded
Multinomial_distribution
Error-correcting codes
subsets, so the algorithm is impractical. The number of subsets is the binomial coefficient, ( n k ) = n ! ( n − k ) ! k ! {\textstyle {\binom {n}{k}}={n! \over
Reed–Solomon_error_correction
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
Concept in combinatorics (part of mathematics)
q-factorials, one can move on to define the q-binomial coefficients, also known as the Gaussian binomial coefficients, as [ n k ] q = [ n ] ! q [ n − k ] ! q
Q-Pochhammer_symbol
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
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
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)
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
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
Product of consecutive factorial numbers
and binomial topics Combinatorial principles Core Addition principle · Rule of product · Binomial coefficient · Pascal's triangle · Central binomial coefficient
Superfactorial
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
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
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
Theorem in p-adic analysis
{\displaystyle n} th binomial coefficient polynomial. Here, the n {\displaystyle n} th forward difference is computed by the binomial transform, so that
Mahler's_theorem
German mathematician (1905–1980)
the power set of an n-set (a Sperner family) is at most the middle binomial coefficient(s). It has several proofs and numerous generalizations, including
Emanuel_Sperner
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
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
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)
Measure of the shape of a function
i}(x-a)^{i}(a-b)^{n-i}} where ( n i ) {\textstyle {\binom {n}{i}}} is the binomial coefficient, it follows that the moments about b can be calculated from the moments
Moment_(mathematics)
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
Theorem on the largest antichain of sets
_{k=0}^{n}{s_{k} \over {n \choose k}}\leq 1,} and multiplying both sides by the binomial coefficient in the denominator of the left hand side gives | S | = ∑ k = 0 n
Sperner's_theorem
French mathematician (1667–1754)
Find the coefficient of the middle term [of a binomial expansion] for a very large and even power [n], or find the ratio that the coefficient of the middle
Abraham_de_Moivre
Expectation or average of the falling factorial of a random variable
involve Stirling numbers of the second kind. If a random variable X has a binomial distribution with success probability p ∈ [0,1] and number of trials n
Factorial_moment
Chances of card combinations in poker
respectively. The Total line also needs adjusting. Games portal Binomial coefficient Combination Combinatorial game theory Effective hand strength algorithm
Poker_probability
Mathematical function
uses a discrete Gaussian kernel, which may be approximated by the Binomial coefficient or sampling a Gaussian. In geostatistics they have been used for
Gaussian_function
Identity involving binomial coefficients, first established by Zhi-Wei Sun in 2002
combinatorics, Sun's curious identity is the following identity involving binomial coefficients, first established by Zhi-Wei Sun in 2002: (x+m+1)∑i=0m(−1)i(x+
Sun's_curious_identity
Polynomial with reversed root positions
P is antipalindromic if P(x)=−P∗(x). From the properties of the binomial coefficients, it follows that the polynomials P(x)=(x+1)n are palindromic for
Reciprocal_polynomial
{\displaystyle k} short notes; this is equivalent to finding the binomial coefficients. The ideas of the Bhagavati were generalized by the Indian mathematician
History_of_combinatorics
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)
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BINOMIAL COEFFICIENT
BINOMIAL COEFFICIENT
BINOMIAL COEFFICIENT
BINOMIAL COEFFICIENT
BINOMIAL COEFFICIENT
BINOMIAL COEFFICIENT
BINOMIAL COEFFICIENT
BINOMIAL COEFFICIENT
BINOMIAL COEFFICIENT
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