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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
Gaussian binomial coefficients (also called Gaussian coefficients, Gaussian polynomials, or q-binomial coefficients) are q-analogs of the binomial coefficients
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
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
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
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
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
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
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)
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)
Describes the highest power of primes dividing a binomial coefficient
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
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
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
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
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
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
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
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
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
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
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
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
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)
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
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
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
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 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
_{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
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
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
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)
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
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
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
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
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
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)
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
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
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
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
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
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
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
Natural number
× 11 × 13, sphenic number, pentagonal number, pentatope number ( binomial coefficient ( 13 4 ) {\displaystyle {\tbinom {13}{4}}} ), Harshad number, member
700_(number)
Cross-validation technique for time series and financial data
Then: The number of unique train-test combinations is given by the binomial coefficient: ( N k ) {\displaystyle {\binom {N}{k}}} Each observation is used
Purged_cross-validation
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)
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
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
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
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
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
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
Discrete analog of a derivative
k ! {\displaystyle {\binom {x}{k}}={\frac {(x)_{k}}{k!}}} is the binomial coefficient, and ( x ) k = x ( x − 1 ) ( x − 2 ) ⋯ ( x − k + 1 ) {\displaystyle
Finite_difference
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
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
of change binomial coefficient Any of the positive integers that occurs as a coefficient in the binomial theorem is a binomial coefficient. Commonly,
Glossary_of_calculus
(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
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
(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
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
Type of mathematical proposition
cannot be directly applied when p = 2 {\displaystyle p=2} because the binomial coefficient ( p 2 ) = p ( p − 1 ) 2 {\displaystyle {\binom {p}{2}}={\frac {p(p-1)}{2}}}
Lifting-the-exponent_lemma
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
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
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
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
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
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
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 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
Polyhedral number representing a tetrahedron
{3}}}{3!}}} The tetrahedral numbers can also be represented as binomial coefficients: T e n = ( n + 2 3 ) . {\displaystyle Te_{n}={\binom {n+2}{3}}.}
Tetrahedral_number
Numerical measure of a statistical relationship between variables
A correlation coefficient is a numerical measure of some type of linear correlation, meaning a linear function between two variables. The variables may
Correlation_coefficient
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
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
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
factorial function. Barnes G-function Beta function: Corresponding binomial coefficient analogue. Digamma function, Polygamma function Incomplete beta function
List of mathematical functions
List_of_mathematical_functions
Rational surface in 5-dimensional projective space
\mathbb {P} ^{m}} with m given by the multiset coefficient, or more familiarly the binomial coefficient, as: m = ( ( n + 1 d ) ) − 1 = ( n + d d ) − 1
Veronese_surface
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
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
Concept in genetics
factorial function. This expression can also be formulated using the binomial coefficient, ( 2 N k ) p k q 2 N − k {\displaystyle {2N \choose k}p^{k}q^{2N-k}}
Genetic_drift
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
Character encoding used in radio data protocols
for 7 bit positions, and the number can be calculated using the binomial coefficient: ( 7 3 ) = ( 7 4 ) = 35 . {\displaystyle \ \textstyle {\binom
CCIR_476
Historical term in mathematics
polynomials. Consider, for example, the ordinary binomial expansion (which contains a binomial coefficient): ( y + x ) n = ∑ k = 0 n ( n k ) y n − k x k
Umbral_calculus
Natural number
0 786 might be the largest n for which the value of the central binomial coefficient 2 n C n {\displaystyle {}_{2n}\!C_{n}} is not divisible by an odd
786_(number)
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 concept named for Ernst Witt
x − 1 ) p = 0 {\displaystyle (x-1)^{p}=0} . By expanding using binomial coefficients, the operation of raising to the p {\displaystyle p} -th power,
Witt_vector
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)
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
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
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
BINOMIAL COEFFICIENT
BINOMIAL COEFFICIENT
BINOMIAL COEFFICIENT
BINOMIAL COEFFICIENT
Girl/Female
English American
Born in the fall; The fall season.
Boy/Male
Hindu, Indian, Kannada
Forehead Eyes Shiva
Boy/Male
Hindu
Young, Lord Krishna
Girl/Female
Greek
Gods incarnate.
Girl/Female
Bengali, Indian, Japanese
Eyes Bright; Eyes; Beauty
Girl/Female
Australian, Greek
A Flower Name; Similar to Samantha
Boy/Male
Greek
A king of Arcadia.
Boy/Male
Hindu, Indian, Marathi
Lord of the Gods
Boy/Male
Hindu, Indian
Diamond
Boy/Male
Australian, Gaelic
Great
BINOMIAL COEFFICIENT
BINOMIAL COEFFICIENT
BINOMIAL COEFFICIENT
BINOMIAL COEFFICIENT
BINOMIAL COEFFICIENT
a.
Consisting of three terms; of or pertaining to trinomials; as, a trinomial root.
a.
Binominal.
n.
A numerical coefficient in any particular case of the binomial theorem.
n.
An invariable quantity; specifically, a function of the coefficients of one or more forms, which remains unaltered, when these undergo suitable linear transformations.
n.
A quantity consisting of three terms, connected by the sign + or -; as, x + y + z, or ax + 2b - c2.
n.
The operation of finding the primitive function which has a given function for its differential coefficient. See Integral.
a.
Of or pertaining to two names; binomial.
n.
A monomial.
a.
Having two names; -- used of the system by which every animal and plant receives two names, the one indicating the genus, the other the species, to which it belongs.
n.
A number or letter put before a letter or quantity, known or unknown, to show how many times the latter is to be taken; as, 6x; bx; here 6 and b are coefficients of x.
n. & a.
Trinomial.
n.
A number, commonly used in computation as a factor, expressing the amount of some change or effect under certain fixed conditions as to temperature, length, volume, etc.; as, the coefficient of expansion; the coefficient of friction.
n.
An expression consisting of two terms connected by the sign plus (+) or minus (-); as, a + b, or 7 - 3.
n.
An expression of the condition of equality between two algebraic quantities or sets of quantities, the sign = being placed between them; as, a binomial equation; a quadratic equation; an algebraic equation; a transcendental equation; an exponential equation; a logarithmic equation; a differential equation, etc.
n.
A name or term.
n.
In the theory of gravitation, or of other forces acting in space, a function of the rectangular coordinates which determine the position of a point, such that its differential coefficients with respect to the coordinates are equal to the components of the force at the point considered; -- also called potential function, or force function. It is called also Newtonian potential when the force is directed to a fixed center and is inversely as the square of the distance from the center.
n.
A rule or principle expressed in algebraic language; as, the binominal formula.
a.
Consisting of two terms; pertaining to binomials; as, a binomial root.
a.
Consisting of but a single term or expression.
n.
A single algebraic expression; that is, an expression unconnected with any other by the sign of addition, substraction, equality, or inequality.