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



*        Evaluate binomial coefficients - 29/09/2015
BINOMIAL CSECT
         USING  BINOMIAL,R15       set base register
         SR     R4,R4              clear for mult and div 
         LA     R5,1               r=1
         LA     R7,1               i=1
         L      R8,N               m=n
LOOP     LR     R4,R7              do while i<=k
         C      R4,K               i<=k
         BH     LOOPEND            if not then exit while
         MR     R4,R8              r*m
         DR     R4,R7              r=r*m/i
         LA     R7,1(R7)           i=i+1
         BCTR   R8,0               m=m-1
         B      LOOP               loop while
LOOPEND  XDECO  R5,PG              edit r
         XPRNT  PG,12              print r
         XR     R15,R15            set return code
         BR     R14                return to caller
N        DC     F'10'              <== input value
K        DC     F'4'               <== input value
PG       DS     CL12               buffer
         YREGS
         END    BINOMIAL