scipy.special.binom#
- scipy.special.binom(x, y, out=None) = <ufunc 'binom'>#
Binomial coefficient considered as a function of two real variables.
For real arguments, the binomial coefficient is defined as
\[\binom{x}{y} = \frac{\Gamma(x + 1)}{\Gamma(y + 1)\Gamma(x - y + 1)} = \frac{1}{(x + 1)\mathrm{B}(x - y + 1, y + 1)}\]Where \(\Gamma\) is the Gamma function (
gamma) and \(\mathrm{B}\) is the Beta function (beta) [1].- Parameters:
- x, y: array_like
Real arguments to \(\binom{x}{y}\).
- outndarray, optional
Optional output array for the function values
- Returns:
- scalar or ndarray
Value of binomial coefficient.
See also
combThe number of combinations of N things taken k at a time.
Notes
The Gamma function has poles at non-positive integers and tends to either positive or negative infinity depending on the direction on the real line from which a pole is approached. When considered as a function of two real variables, \(\binom{x}{y}\) is thus undefined when x is a negative integer.
binomreturnsnanwhenxis a negative integer. This is the case even whenxis a negative integer andyan integer, contrary to the usual convention for defining \(\binom{n}{k}\) when it is considered as a function of two integer variables.References
Examples
The following examples illustrate the ways in which
binomdiffers from the functioncomb.>>> from scipy.special import binom, comb
When
exact=Falseandxandyare both positive,combcallsbinominternally.>>> x, y = 3, 2 >>> (binom(x, y), comb(x, y), comb(x, y, exact=True)) (3.0, 3.0, 3)
For larger values,
combwithexact=Trueno longer agrees withbinom.>>> x, y = 43, 23 >>> (binom(x, y), comb(x, y), comb(x, y, exact=True)) (960566918219.9999, 960566918219.9999, 960566918220)
binomreturnsnanwhenxis a negative integer, but is otherwise defined for negative arguments.combreturns 0 whenever one ofxoryis negative orxis less thany.>>> x, y = -3, 2 >>> (binom(x, y), comb(x, y)) (nan, 0.0)
>>> x, y = -3.1, 2.2 >>> (binom(x, y), comb(x, y)) (18.714147876804432, 0.0)
>>> x, y = 2.2, 3.1 >>> (binom(x, y), comb(x, y)) (0.037399983365134115, 0.0)