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

comb

The 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. binom returns nan when x is a negative integer. This is the case even when x is a negative integer and y an integer, contrary to the usual convention for defining \(\binom{n}{k}\) when it is considered as a function of two integer variables.

binom has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable SCIPY_ARRAY_API=1 and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

Library

CPU

GPU

NumPy

✅

n/a

CuPy

n/a

✅

PyTorch

✅

⛔

JAX

⚠️ no JIT

⛔

Dask

✅

n/a

See Support for the array API standard for more information.

References

Examples

The following examples illustrate the ways in which binom differs from the function comb.

>>> from scipy.special import binom, comb

When exact=False and x and y are both positive, comb calls binom internally.

>>> x, y = 3, 2
>>> (binom(x, y), comb(x, y), comb(x, y, exact=True))
(3.0, 3.0, 3)

For larger values, comb with exact=True no longer agrees with binom.

>>> x, y = 43, 23
>>> (binom(x, y), comb(x, y), comb(x, y, exact=True))
(960566918219.9999, 960566918219.9999, 960566918220)

binom returns nan when x is a negative integer, but is otherwise defined for negative arguments. comb returns 0 whenever one of x or y is negative or x is less than y.

>>> 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)
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