BUILT-IN MATHEMATICA SYMBOL

Binomial

gives the binomial coefficient

.

MORE INFORMATION

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.

EXAMPLES

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Basic Examples (2)

Evaluate 10 choose 3:

Construct Pascal's triangle:

Evaluate 10 choose 3:

In[1]:=

Out[1]=

Construct Pascal's triangle:

In[1]:=

Out[1]=

Scope (8)

Evaluate special cases for symbolic arguments:

Evaluate Binomial for large arguments:

Evaluate for half-integer arguments:

Evaluate numerically:

Evaluate for complex arguments:

Compute sums involving Binomial:

Series expansion at a generic point:

TraditionalForm formatting:

Generalizations & Extensions (2)

Infinite arguments give symbolic results:

Binomial threads element-wise over lists:

Applications (8)

Illustrate the binomial theorem:

Fractional binomial theorem:

Binomial coefficients mod 2:

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 (4)

Use FullSimplify to simplify expressions involving binomial coefficients:

Use FunctionExpand to expand into Gamma functions:

Sums involving Binomial:

Find the generating function 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:

Neat Examples (6)

Binomial coefficient mod

:

Closed-form inverse of Hilbert matrices:

Nested binomials over the complex plane:

Plot Binomial at infinity:

Plot Binomial for complex arguments:

Plot Binomial at Gaussian integers: