This programming task, is to calculate ANY binomial coefficient.

However, it has to be able to output   






(


5
3


)





{\displaystyle {\binom {5}{3}}}

,   which is   10.

This formula is recommended:


 






(


n
k


)



=



n
!


(
n
−
k
)
!
k
!



=



n
(
n
−
1
)
(
n
−
2
)
…
(
n
−
k
+
1
)


k
(
k
−
1
)
(
k
−
2
)
…
1





{\displaystyle {\binom {n}{k}}={\frac {n!}{(n-k)!k!}}={\frac {n(n-1)(n-2)\ldots (n-k+1)}{k(k-1)(k-2)\ldots 1}}}





See Also:

 Combinations and permutations
 Pascal's triangle

 The number of samples of size k from n objects.


With   combinations and permutations   generation tasks. 






 Order Unimportant

 Order Important


 Without replacement








(


n
k


)




=

n



C

k


=



n
(
n
−
1
)
…
(
n
−
k
+
1
)


k
(
k
−
1
)
…
1





{\displaystyle {\binom {n}{k}}=^{n}\operatorname {C} _{k}={\frac {n(n-1)\ldots (n-k+1)}{k(k-1)\dots 1}}}










n



P

k


=
n
⋅
(
n
−
1
)
⋅
(
n
−
2
)
⋯
(
n
−
k
+
1
)


{\displaystyle ^{n}\operatorname {P} _{k}=n\cdot (n-1)\cdot (n-2)\cdots (n-k+1)}




Task: Combinations

Task: Permutations


 With replacement








(



n
+
k
−
1

k


)




=

n
+
k
−
1



C

k


=



(
n
+
k
−
1
)
!


(
n
−
1
)
!
k
!





{\displaystyle {\binom {n+k-1}{k}}=^{n+k-1}\operatorname {C} _{k}={(n+k-1)! \over (n-1)!k!}}








n

k




{\displaystyle n^{k}}




Task: Combinations with repetitions

Task: Permutations with repetitions



using System;
 
namespace BinomialCoefficients
{
    class Program
    {
        static void Main(string[] args)
        {
            ulong n = 1000000, k = 3;
            ulong result = biCoefficient(n, k);
            Console.WriteLine("The Binomial Coefficient of {0}, and {1}, is equal to: {2}", n, k, result);
            Console.ReadLine();
        }
 
        static int fact(int n)
        {
            if (n == 0) return 1;
            else return n * fact(n - 1);
        }
 
        static ulong biCoefficient(ulong n, ulong k)
        {
            if (k > n - k)
            {
                k = n - k;
            }
 
            ulong c = 1;
            for (uint i = 0; i < k; i++)
            {
                c = c * (n - i);
                c = c / (i + 1);
            }
            return c;
        }
    }
}