#include <boost/math/special_functions/binomial.hpp>

namespace boost{ namespace math{

template <class T>
T binomial_coefficient(unsigned n, unsigned k);

template <class T, class Policy>
T binomial_coefficient(unsigned n, unsigned k, const Policy&);

}} // namespaces

Returns the binomial coefficient: _nC_k.

Requires k <= n.

The final Policy argument is optional and can be used to control the behaviour of the function: how it handles errors, what level of precision to use etc. Refer to the policy documentation for more details.

May return the result of overflow_error if the result is too large to represent in type T.

Accuracy

The accuracy will be the same as for the factorials for small arguments (i.e. no more than one or two epsilon), and the beta function for larger arguments.

Testing

The spot tests for the binomial coefficients use data generated by functions.wolfram.com.

Implementation

Binomial coefficients are calculated using table lookup of factorials where possible using:

_nC_k = n! / (k!(n-k)!)

Otherwise it is implemented in terms of the beta function using the relations:

_nC_k = 1 / (k * beta(k, n-k+1))

and

_nC_k = 1 / ((n-k) * beta(k+1, n-k))