binomial coefficient gamma

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Computes the binomial coefficient with the given arguments.

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Binomial Coefficient Gamma
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Interface

C++

Excel

Binomial Coefficient Gamma

intbinomial_coefficient_gamma(	int	`n`
	int	`k`	)

This function calculates the binomial coefficient $\inline C(n, k)$ directly, using the logarithm of the Gamma function, rather than recursively. $\inline C(n, k)$ is the number of distinct combinations of k objects chosen from a set of n distinct objects. A combination is like a set, in that order does not matter. The computed expression has the following form:

$C(n, k) = \left( \begin{array}{c} n \cr k \end{array} \right) = \frac{n!}{k!(n - k)!}$

(2)

This coefficient represents the number of distinct combinations of k objects chosen from a set of n distinct objects. The advantage in using this function is that large values of n and k may be given, without the fear of arithmetic overflow during the intermediate calculations. This is possible with the use of the logarithm of the Gamma function.

Example:

#include <codecogs/maths/combinatorics/arithmetic/binomial_coefficient_gamma.h>
#include <iostream>
int main()
{
  for (int i = 0; i <= 6; i++)
  {
    std::cout << "C(6, " << i << ") = ";
    std::cout << Maths::Combinatorics::Arithmetic::binomial_coefficient_gamma(6, i);
    std::cout << std::endl;
  }
  return 0;
}

Output:

C(6, 0) = 1
C(6, 1) = 6
C(6, 2) = 15
C(6, 3) = 20
C(6, 4) = 15
C(6, 5) = 6
C(6, 6) = 1

References:

SUBSET, a C++ library of combinatorial routines, http://www.csit.fsu.edu/~burkardt/cpp_src/subset/subset.html

Parameters

n	the first combinatorial parameter
k	the second combinatorial parameter

Returns

the binomial coefficients with parameters n and k

Authors

Lucian Bentea (August 2005)

Source Code

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