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Binomial
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Binomial

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Binomial[n,m]

gives the binomial coefficient TemplateBox[{n, m}, Binomial].

Details

  • Integer mathematical function, suitable for both symbolic and numerical manipulation.
  • Binomial is also known as combinations and as choose function.
  • Binomial gives the symmetric coefficients for negative integer . Use PascalBinomial for coefficients that preserve Pascal's identity for all integer values. Binomial and PascalBinomial agree except for negative integer .
  • In general, TemplateBox[{n, m}, Binomial] is defined by (n!)/(m!(n-m)!)=(TemplateBox[{{n, +, 1}}, Gamma])/(TemplateBox[{{m, +, 1}}, Gamma] TemplateBox[{{n, -, m, +, 1}}, Gamma]) or suitable limits of this.
  • When is a negative integer, TemplateBox[{n, m}, Binomial]=TemplateBox[{{{(, TemplateBox[{{n, +, epsilon, +, 1}}, Gamma], )}, /, {(, {TemplateBox[{{m, +, epsilon, +, 1}}, Gamma], , TemplateBox[{{{-, m}, +, n, +, epsilon, +, 1}}, Gamma]}, )}}, epsilon, 0, TemplateBox[{}, Complexes]}, LimitWithTooltip]. »
  • The particular limit chosen preserves the symmetry rule TemplateBox[{n, m}, Binomial]=TemplateBox[{n, {n, -, m}}, Binomial] for all complex and . »
  • Pascal's identity TemplateBox[{n, m}, Binomial]=TemplateBox[{{n, -, 1}, m}, Binomial]+TemplateBox[{{n, -, 1}, {m, -, 1}}, Binomial] is satisfied for almost all and , but violated for . »
  • 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 TemplateBox[{n, k}, Binomial] of and . For non-negative integers and , the binomial coefficient has value TemplateBox[{n, k}, Binomial]=(n!)/(k! (n-k)!), where is the Factorial function. By symmetry, TemplateBox[{n, k}, Binomial]=TemplateBox[{n, {n, -, k}}, Binomial]. The binomial coefficient is important in probability theory and combinatorics and is sometimes also denoted
  • For non-negative integers and , the binomial coefficient TemplateBox[{n, k}, Binomial] 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 , (x+y)^n=sum_(k=0)^nTemplateBox[{n, k}, Binomial] x^(n-k)y^k. 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 TemplateBox[{n, k}, Binomial]=TemplateBox[{{n, -, 1}, {k, -, 1}}, Binomial]+TemplateBox[{{n, -, 1}, k}, Binomial].
  • Expressing factorials as gamma functions generalizes the binomial coefficient to complex and as TemplateBox[{x, y}, Binomial]=(TemplateBox[{{x, +, 1}}, Gamma])/(TemplateBox[{{y, +, 1}}, Gamma] TemplateBox[{{x, -, y, +, 1}}, Gamma]). Using the symmetry formula (TemplateBox[{{s, -, a, +, 1}}, Gamma])/(TemplateBox[{{s, -, b, +, 1}}, Gamma])=(-1)^(b-a)( TemplateBox[{{b, -, s}}, Gamma])/(TemplateBox[{{a, -, s}}, Gamma]) 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 (-1)^k TemplateBox[{{k, -, n, -, 1}, k}, Binomial] if , (-1)^(n-k) TemplateBox[{{{-, k}, -, 1}, {n, -, k}}, Binomial] 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 TemplateBox[{n, k}, Binomial] is the multinomial coefficient (n;k,n-k).

Examples

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Basic Examples  (5)Summary of the most common use cases

Evaluate numerically:

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In[1]:=1
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https://wolfram.com/xid/0cq0qv3e-exyjo0
Direct link to example
Out[1]=1

Evaluate symbolically:

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In[1]:=1
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https://wolfram.com/xid/0cq0qv3e-34wfwl
Direct link to example
Out[1]=1

Construct Pascal's triangle:

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In[1]:=1
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https://wolfram.com/xid/0cq0qv3e-omwgp3
Direct link to example
Out[1]=1

Plot over a subset of the reals as a function of its first parameter:

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In[1]:=1
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https://wolfram.com/xid/0cq0qv3e-gi13sj
Direct link to example
Out[1]=1

Plot over a subset of the reals as a function of its second parameter:

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In[2]:=2
✖
https://wolfram.com/xid/0cq0qv3e-kvw8g
Direct link to example
Out[2]=2

Plot over a subset of the complexes:

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In[1]:=1
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https://wolfram.com/xid/0cq0qv3e-kiedlx
Direct link to example
Out[1]=1

Scope  (35)Survey of the scope of standard use cases

Numerical Evaluation  (7)

Evaluate numerically:

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In[1]:=1
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https://wolfram.com/xid/0cq0qv3e-l274ju
Direct link to example
Out[1]=1
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In[2]:=2
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https://wolfram.com/xid/0cq0qv3e-cksbl4
Direct link to example
Out[2]=2

Evaluate for half-integer arguments:

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In[1]:=1
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https://wolfram.com/xid/0cq0qv3e-klrel
Direct link to example
Out[1]=1

Evaluate to high precision:

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e-b0wt9
Direct link to example
Out[1]=1

The precision of the output tracks the precision of the input:

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In[2]:=2
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https://wolfram.com/xid/0cq0qv3e-y7k4a
Direct link to example
Out[2]=2

Complex number inputs:

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In[1]:=1
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https://wolfram.com/xid/0cq0qv3e-c3zszs
Direct link to example
Out[1]=1
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In[2]:=2
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https://wolfram.com/xid/0cq0qv3e-hfml09
Direct link to example
Out[2]=2

Evaluate efficiently at high precision:

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In[1]:=1
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https://wolfram.com/xid/0cq0qv3e-di5gcr
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0cq0qv3e-bq2c6r
Direct link to example
Out[2]=2

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

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In[1]:=1
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https://wolfram.com/xid/0cq0qv3e-h0d6g
Direct link to example
Out[1]=1
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In[2]:=2
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https://wolfram.com/xid/0cq0qv3e-dj6d9x
Direct link to example
Out[2]=2

Or compute average-case statistical intervals using Around:

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In[3]:=3
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https://wolfram.com/xid/0cq0qv3e-cw18bq
Direct link to example
Out[3]=3

Compute the elementwise values of an array:

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In[1]:=1
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https://wolfram.com/xid/0cq0qv3e-thgd2
Direct link to example
Out[1]=1

Or compute the matrix Binomial function using MatrixFunction:

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In[2]:=2
✖
https://wolfram.com/xid/0cq0qv3e-o5jpo
Direct link to example
Out[2]=2

Specific Values  (4)

Values of Binomial at particular points:

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In[1]:=1
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https://wolfram.com/xid/0cq0qv3e-nww7l
Direct link to example
Out[1]=1

Binomial for symbolic n:

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In[1]:=1
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https://wolfram.com/xid/0cq0qv3e-fc9m8o
Direct link to example
Out[1]=1

Values at zero:

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In[1]:=1
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https://wolfram.com/xid/0cq0qv3e-jdd1mk
Direct link to example
Out[1]=1
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In[2]:=2
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https://wolfram.com/xid/0cq0qv3e-e41pf2
Direct link to example
Out[2]=2

Note that this is zero on all integers away from :

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In[3]:=3
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https://wolfram.com/xid/0cq0qv3e-1ys7jj
Direct link to example
Out[3]=3

Find a value of n for which Binomial[n,2]=15:

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In[1]:=1
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https://wolfram.com/xid/0cq0qv3e-f2hrld
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0cq0qv3e-gxsukt
Direct link to example
Out[2]=2

Visualization  (3)

Plot the Binomial as a function of its parameter n:

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In[1]:=1
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https://wolfram.com/xid/0cq0qv3e-bhh92c
Direct link to example
Out[1]=1

Plot the Binomial as a function of its parameter :

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e-bzo8ib
Direct link to example
Out[1]=1

Plot the real part of TemplateBox[{z, 5}, Binomial]:

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e-cjk9wl
Direct link to example
Out[1]=1

Plot the imaginary part of TemplateBox[{z, 5}, Binomial]:

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In[2]:=2
✖
https://wolfram.com/xid/0cq0qv3e-b41shq
Direct link to example
Out[2]=2

Function Properties  (12)

Real domain of Binomial as a function of its parameter n:

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e-cl7ele
Direct link to example
Out[1]=1

Real domain of Binomial as a function of its parameter m:

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In[2]:=2
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https://wolfram.com/xid/0cq0qv3e-bhcc
Direct link to example
Out[2]=2

Complex domains:

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In[3]:=3
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https://wolfram.com/xid/0cq0qv3e-de3irc
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0cq0qv3e-d1f4tb
Direct link to example
Out[4]=4

Function range of Binomial:

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In[1]:=1
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https://wolfram.com/xid/0cq0qv3e-d8uhh9
Direct link to example
Out[1]=1

Binomial has the mirror property TemplateBox[{TemplateBox[{z}, Conjugate, SyntaxForm -> SuperscriptBox], 2}, Binomial]=TemplateBox[{TemplateBox[{z, 2}, Binomial]}, Conjugate]:

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e-heoddu
Direct link to example
Out[1]=1

Compute sums involving Binomial:

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e-m4i2hj
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0cq0qv3e-d5f73c
Direct link to example
Out[2]=2

When is positive, TemplateBox[{x, y}, Binomial] is an analytic function of both variables:

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e-g1bb8w
Direct link to example
Out[1]=1

This is not true for negative :

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In[2]:=2
✖
https://wolfram.com/xid/0cq0qv3e-yoc2vi
Direct link to example
Out[2]=2

TemplateBox[{x, 7}, Binomial] is neither non-decreasing nor non-increasing:

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e-2ra8g
Direct link to example
Out[1]=1

TemplateBox[{x, 7}, Binomial] is not injective:

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e-c9npzh
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0cq0qv3e-b5buvp
Direct link to example
Out[2]=2

TemplateBox[{x, 7}, Binomial] is surjective:

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e-patce
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0cq0qv3e-bcrbvs
Direct link to example
Out[2]=2

Binomial is neither non-negative nor non-positive:

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e-dvzykj
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0cq0qv3e-kpi9jh
Direct link to example
Out[2]=2

TemplateBox[{x, y}, Binomial] has singularities and discontinuities where is a negative integer:

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e-tizf8k
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0cq0qv3e-g7sk2p
Direct link to example
Out[2]=2

TemplateBox[{x, 7}, Binomial] is neither convex nor concave:

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e-l0srvu
Direct link to example
Out[1]=1

TraditionalForm formatting:

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e-zvmzu
Direct link to example

Differentiation  (3)

First derivative with respect to :

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e-krpoah
Direct link to example
Out[1]=1

Higher derivatives with respect to :

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e-z33jv
Direct link to example
Out[1]=1

Plot the higher derivatives with respect to for :

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In[2]:=2
✖
https://wolfram.com/xid/0cq0qv3e-fxwmfc
Direct link to example
Out[2]=2

First derivative with respect to :

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e-pgq6et
Direct link to example
Out[1]=1

Series Expansions  (4)

Find the Taylor expansion using Series:

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e-ewr1h8
Direct link to example
Out[1]=1

Plots of the first three approximations around :

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e-binhar
Direct link to example
Out[2]=2

Find the series expansion at Infinity:

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e-ht5z5a
Direct link to example
Out[1]=1

Find series expansion for an arbitrary symbolic direction :

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e-rkoq2k
Direct link to example
Out[1]=1

Taylor expansion at a generic point:

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e-jwxla7
Direct link to example
Out[1]=1

Function Identities and Simplifications  (2)

Functional identity:

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e-fntli3
Direct link to example
Out[1]=1

Recurrence relations:

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e-ge3pv7
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0cq0qv3e-c93h2q
Direct link to example
Out[2]=2

Generalizations & Extensions  (2)Generalized and extended use cases

Infinite arguments give symbolic results:

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0cq0qv3e
Direct link to example
Out[2]=2

Binomial threads elementwise over lists:

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e
Direct link to example
Out[1]=1

Applications  (11)Sample problems that can be solved with this function

There are TemplateBox[{n, m}, Binomial] ways to choose elements without replacements from a set of elements:

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e-bzis8
Direct link to example
Out[1]=1

Check with direct enumeration:

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In[2]:=2
✖
https://wolfram.com/xid/0cq0qv3e-drjmvc
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0cq0qv3e-bp5vcd
Direct link to example
Out[3]=3

There are TemplateBox[{{m, +, n, -, 1}, m}, Binomial] ways to choose elements with replacement from a set of elements:

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e-basc4x
Direct link to example
Out[1]=1

Check with direct enumeration:

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In[2]:=2
✖
https://wolfram.com/xid/0cq0qv3e-eck30t
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0cq0qv3e-fqjyd5
Direct link to example
Out[3]=3

There are TemplateBox[{{m, +, n}, m}, Binomial] ways to arrange indistinguishable objects of one kind, and indistinguishable objects of another kind:

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e-bc420f
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0cq0qv3e-zyhq5
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0cq0qv3e-oqvidf
Direct link to example
Out[3]=3

Illustrate the binomial theorem:

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e-elh4b
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0cq0qv3e-ddn5wi
Direct link to example
Out[2]=2

Fractional binomial theorem:

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e-bybnro
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0cq0qv3e-cxa6k4
Direct link to example
Out[2]=2

Binomial coefficients mod 2:

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e-mczzkd
Direct link to example
Out[1]=1

Plot Binomial in the arguments' plane:

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e-ig68fq
Direct link to example
Out[1]=1

Plot the logarithm of the number of ways to pick elements out of :

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e-j6x0v6
Direct link to example
Out[1]=1

Compute higher derivatives of a product of two functions:

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e-dq0poq
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0cq0qv3e
Direct link to example
Out[2]=2

PDF of the binomial probability distribution:

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0cq0qv3e
Direct link to example
Out[2]=2

Bernstein polynomials are defined in terms of Binomial:

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e-ctuozw
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0cq0qv3e-4cmm9
Direct link to example
Out[2]=2

Properties & Relations  (11)Properties of the function, and connections to other functions

On the integers, Binomial[n,m] equals TemplateBox[{{{(, TemplateBox[{{n, +, epsilon, +, 1}}, Gamma], )}, /, {(, {TemplateBox[{{m, +, epsilon, +, 1}}, Gamma], , TemplateBox[{{{-, m}, +, n, +, epsilon, +, 1}}, Gamma]}, )}}, epsilon, 0, TemplateBox[{}, Complexes]}, LimitWithTooltip]:

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e-k4fwj3
Direct link to example
Out[1]=1

This can be expressed as (-1)^m TemplateBox[{{-, n}, m}, Pochhammer]/m! for and (-1)^(n-m) TemplateBox[{{-, n}, {n, -, m}}, Pochhammer]/(n-m)! for :

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In[2]:=2
✖
https://wolfram.com/xid/0cq0qv3e-zru3f9
Direct link to example
Out[2]=2

An alternative formula on the integers:

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In[3]:=3
✖
https://wolfram.com/xid/0cq0qv3e-fnhbix
Direct link to example
Out[3]=3

Pascal's identity is satisfied almost everywhere:

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e-x2cynh
Direct link to example
Out[1]=1

It is violated at the origin:

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In[2]:=2
✖
https://wolfram.com/xid/0cq0qv3e-jkv9k6
Direct link to example
Out[2]=2

PascalBinomial satisfies the identity everywhere, including the origin:

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In[3]:=3
✖
https://wolfram.com/xid/0cq0qv3e-wasx40
Direct link to example
Out[3]=3

The symmetry rule TemplateBox[{n, m}, PascalBinomial]=TemplateBox[{n, {n, -, m}}, PascalBinomial] holds for all values of and :

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e-i67mi5
Direct link to example
Out[1]=1

PascalBinomial performs simple evaluations for symbolic arguments:

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e-jf1l2z
Direct link to example
Out[1]=1

For more complex expressions, it will avoid automatic expansion:

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In[2]:=2
✖
https://wolfram.com/xid/0cq0qv3e-4m4kxv
Direct link to example
Out[2]=2

Use FunctionExpand with conditions to achieve appropriate simplifications:

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In[3]:=3
✖
https://wolfram.com/xid/0cq0qv3e-ty4v8l
Direct link to example
Out[3]=3

Use FullSimplify to simplify expressions involving binomial coefficients:

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e-ctylci
Direct link to example
Out[1]=1

Use FunctionExpand to expand into Gamma functions:

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e
Direct link to example
Out[1]=1

Sums involving Binomial:

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0cq0qv3e
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0cq0qv3e
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0cq0qv3e
Direct link to example
Out[4]=4

Find the generating function Binomial:

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e
Direct link to example
Out[1]=1

Binomial can be represented as a DifferenceRoot:

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e-d7mw27
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0cq0qv3e-ojcks5
Direct link to example
Out[2]=2

The generating function for Binomial:

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e-pz93yz
Direct link to example
Out[1]=1

The exponential generating function for Binomial:

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e-g1j8nf
Direct link to example
Out[1]=1

Possible Issues  (3)Common pitfalls and unexpected behavior

Large arguments can give results too large to be computed explicitly:

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e
Direct link to example
Out[1]=1

Machine-number inputs can give high‐precision results:

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0cq0qv3e
Direct link to example
Out[2]=2

As a bivariate function, Binomial is not continuous in both variables at negative integers:

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e-oluz92
Direct link to example
Out[1]=1

The value of Binomial at negative integers is determined via Binomial[n,m]Binomial[n,n-m]:

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In[2]:=2
✖
https://wolfram.com/xid/0cq0qv3e
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0cq0qv3e-ec2vnx
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0cq0qv3e-kkw2gg
Direct link to example
Out[4]=4

Neat Examples  (7)Surprising or curious use cases

Construct a graphical version of Pascal's triangle:

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e-q15ovm
Direct link to example
Out[1]=1

Extend the triangle to negative integers; unlabeled points indicate a zero value:

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In[8]:=8
✖
https://wolfram.com/xid/0cq0qv3e-hx7umr
Direct link to example
Out[8]=8

PascalBinomial, by contrast, zeroes out the top-left sector where both inputs are negative:

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In[9]:=9
✖
https://wolfram.com/xid/0cq0qv3e-sdo9vl
Direct link to example
Out[9]=9

Binomial coefficient mod :

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e-blpqjq
Direct link to example
Out[1]=1

Closed‐form inverse of Hilbert matrices:

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e-mq2uki
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0cq0qv3e-f9gupa
Direct link to example
Out[2]=2

Nested binomials over the complex plane:

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e
Direct link to example
Out[1]=1

Plot Binomial at infinity:

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e
Direct link to example
Out[1]=1

Plot Binomial for complex arguments:

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e
Direct link to example
Out[1]=1

Plot Binomial at Gaussian integers:

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In[1]:=1
✖
https://wolfram.com/xid/0cq0qv3e
Direct link to example
Out[1]=1
Wolfram Research (1988), Binomial, Wolfram Language function, https://reference.wolfram.com/language/ref/Binomial.html.
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✖
Wolfram Research (1988), Binomial, Wolfram Language function, https://reference.wolfram.com/language/ref/Binomial.html.

Text

Wolfram Research (1988), Binomial, Wolfram Language function, https://reference.wolfram.com/language/ref/Binomial.html.

Copy to clipboard.
✖
Wolfram Research (1988), Binomial, Wolfram Language function, https://reference.wolfram.com/language/ref/Binomial.html.

CMS

Wolfram Language. 1988. "Binomial." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Binomial.html.

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✖
Wolfram Language. 1988. "Binomial." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Binomial.html.

APA

Wolfram Language. (1988). Binomial. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Binomial.html

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✖
Wolfram Language. (1988). Binomial. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Binomial.html

BibTeX

@misc{reference.wolfram_2025_binomial, author="Wolfram Research", title="{Binomial}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/Binomial.html}", note=[Accessed: 25-March-2025 ]}

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✖
@misc{reference.wolfram_2025_binomial, author="Wolfram Research", title="{Binomial}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/Binomial.html}", note=[Accessed: 25-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_binomial, organization={Wolfram Research}, title={Binomial}, year={1988}, url={https://reference.wolfram.com/language/ref/Binomial.html}, note=[Accessed: 25-March-2025 ]}

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✖
@online{reference.wolfram_2025_binomial, organization={Wolfram Research}, title={Binomial}, year={1988}, url={https://reference.wolfram.com/language/ref/Binomial.html}, note=[Accessed: 25-March-2025 ]}