gives the binomial coefficient .
- Integer mathematical function, suitable for both symbolic and numerical manipulation.
- In general, is defined by or suitable limits of this.
- For integers and certain other special arguments, Binomial automatically evaluates to exact values.
- Binomial is automatically evaluated symbolically for simple cases; FunctionExpand gives results for other cases.
- Binomial can be evaluated to arbitrary numerical precision.
- Binomial automatically threads over lists.
- Binomial can be used with Interval and CenteredInterval objects. »
Background & Context
- Binomial represents the binomial coefficient function, which returns the binomial coefficient of and . For non-negative integers and , the binomial coefficient has value , where is the Factorial function. By symmetry, . The binomial coefficient is important in probability theory and combinatorics and is sometimes also denoted
- For non-negative integers and , the binomial coefficient gives the number of subsets of length contained in the set . This is also the number of distinct ways of picking elements (without replacement and ignoring order) from the first positive integers and for this reason is often voiced as " choose ".
- The binomial coefficient lies at the heart of the binomial formula, which states that for any non-negative integer , . This interpretation of binomial coefficients is related to the binomial distribution of probability theory, implemented via BinomialDistribution. Another important application is in the combinatorial identity known as Pascal's rule, which relates the binomial coefficient with shifted arguments according to .
- Expressing factorials as gamma functions generalizes the binomial coefficient to complex and as . Using the symmetry formula for integer and and complex then allows the definition of the binomial coefficient to be extended to negative integer arguments, making it continuous at all integer arguments as well as continuous for all complex arguments except for negative integer and noninteger (in which case it is infinite). This definition for negative and integer , given by if , if and 0 otherwise, is in agreement with both the binomial theorem and most combinatorial identities (with a few special exceptions).
- Binomial coefficients are generalized by multinomial coefficients. Multinomial returns the multinomial coefficient (n;n1,…,nk) of given numbers n1,…,nk summing to , where . The binomial coefficient is the multinomial coefficient (n;k,n-k).
Examplesopen allclose all
Basic Examples (6)
Plot over a subset of the reals as a function of its first parameter:
Plot over a subset of the reals as a function of its second parameter:
Plot over a subset of the complexes:
Series expansion at the origin:
Series expansion at Infinity:
Numerical Evaluation (6)
Evaluate for half-integer arguments:
The precision of the output tracks the precision of the input:
Evaluate efficiently at high precision:
Binomial can be used with Interval and CenteredInterval objects:
Specific Values (4)
Function Properties (12)
Real domain of Binomial as a function of its parameter n:
Real domain of Binomial as a function of its parameter m:
Function range of Binomial:
Binomial has the mirror property :
Compute sums involving Binomial:
For fixed, positive , is an analytic function of :
is neither non-decreasing nor non-increasing:
Binomial[x,3] is not injective:
Binomial[x,3] is surjective:
Binomial is neither non-negative nor non-positive:
has singularities and discontinuities where is a negative integer:
is neither convex nor concave:
Series Expansions (4)
Generalizations & Extensions (2)
Infinite arguments give symbolic results:
Binomial threads elementwise over lists:
There are ways to choose elements without replacements from a set of elements:
Check with direct enumeration:
There are ways to choose elements with replacement from a set of elements:
Check with direct enumeration:
There are ways to arrange indistinguishable objects of one kind, and indistinguishable objects of another kind:
Illustrate the binomial theorem:
Plot Binomial in the arguments' plane:
Plot the logarithm of the number of ways to pick elements out of :
Compute higher derivatives of a product of two functions:
PDF of the binomial probability distribution:
Bernstein polynomials are defined in terms of Binomial:
Properties & Relations (7)
Use FullSimplify to simplify expressions involving binomial coefficients:
Use FunctionExpand to expand into Gamma functions:
Sums involving Binomial:
Find the generating function Binomial:
Binomial can be represented as a DifferenceRoot:
The generating function for Binomial:
The exponential generating function for Binomial:
Possible Issues (3)
Large arguments can give results too large to be computed explicitly:
Machine-number inputs can give high‐precision results:
As a bivariate function, Binomial is not continuous in both variables at negative integers:
The value of Binomial at negative integers is determined via Binomial[n,m]Binomial[n,n-m]:
Wolfram Research (1988), Binomial, Wolfram Language function, https://reference.wolfram.com/language/ref/Binomial.html.
Wolfram Language. 1988. "Binomial." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Binomial.html.
Wolfram Language. (1988). Binomial. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Binomial.html