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



#include <stdio.h>
#include <limits.h>
 
/* We go to some effort to handle overflow situations */
 
static unsigned long gcd_ui(unsigned long x, unsigned long y) {
  unsigned long t;
  if (y < x) { t = x; x = y; y = t; }
  while (y > 0) {
    t = y;  y = x % y;  x = t;  /* y1 <- x0 % y0 ; x1 <- y0 */
  }
  return x;
}
 
unsigned long binomial(unsigned long n, unsigned long k) {
  unsigned long d, g, r = 1;
  if (k == 0) return 1;
  if (k == 1) return n;
  if (k >= n) return (k == n);
  if (k > n/2) k = n-k;
  for (d = 1; d <= k; d++) {
    if (r >= ULONG_MAX/n) {  /* Possible overflow */
      unsigned long nr, dr;  /* reduced numerator / denominator */
      g = gcd_ui(n, d);  nr = n/g;  dr = d/g;
      g = gcd_ui(r, dr);  r = r/g;  dr = dr/g;
      if (r >= ULONG_MAX/nr) return 0;  /* Unavoidable overflow */
      r *= nr;
      r /= dr;
      n--;
    } else {
      r *= n--;
      r /= d;
    }
  }
  return r;
}
 
int main() {
    printf("%lu\n", binomial(5, 3));
    printf("%lu\n", binomial(40, 19));
    printf("%lu\n", binomial(67, 31));
    return 0;
}