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



 
10 REM BINOMIAL COEFFICIENTS
20 REM COMMODORE BASIC 2.0
30 REM 2021-08-24
40 REM BY ALVALONGO
100 Z=0:U=1
110 FOR N=U TO 10
120 PRINT N;
130 FOR K=Z TO N
140 GOSUB 900
150 PRINT C;
160 NEXT K
170 PRINT
180 NEXT N
190 END
900 REM BINOMIAL COEFFICIENT
910 IF K<Z OR K>N THEN C=Z:RETURN
920 IF K=Z OR K=N THEN C=U:RETURN
930 P=K:IF N-K<P THEN P=N-K
940 C=U
950 FOR I=Z TO P-U
960 C=C/(I+U)*(N-I)
980 NEXT I
990 RETURN
 