This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# Binomial

 Binomialgives 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.
Evaluate 10 choose 3:
Construct Pascal's triangle:
Evaluate 10 choose 3:
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Construct Pascal's triangle:
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 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:
Infinite arguments give symbolic results:
 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:
Use FullSimplify to simplify expressions involving binomial coefficients:
Use FunctionExpand to expand into Gamma functions:
Sums involving Binomial:
Find the generating function Binomial:
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:
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:
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