Task

Given non-negative integers    m    and    n,   generate all size    m    combinations   of the integers from    0   (zero)   to    n-1    in sorted order   (each combination is sorted and the entire table is sorted).



Example

3   comb    5      is:

0 1 2
0 1 3
0 1 4
0 2 3
0 2 4
0 3 4
1 2 3
1 2 4
1 3 4
2 3 4

If it is more "natural" in your language to start counting from    1   (unity) instead of    0   (zero),

the combinations can be of the integers from    1   to    n. 



See also


 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



with Ada.Text_IO;  use Ada.Text_IO;
 
procedure Test_Combinations is
   generic
      type Integers is range <>;
   package Combinations is
      type Combination is array (Positive range <>) of Integers;
      procedure First (X : in out Combination);
      procedure Next (X : in out Combination); 
      procedure Put (X : Combination);
   end Combinations;
 
   package body Combinations is
      procedure First (X : in out Combination) is
      begin
         X (1) := Integers'First;
         for I in 2..X'Last loop
            X (I) := X (I - 1) + 1;
         end loop;
      end First;
      procedure Next (X : in out Combination) is
      begin
         for I in reverse X'Range loop
            if X (I) < Integers'Val (Integers'Pos (Integers'Last) - X'Last + I) then
               X (I) := X (I) + 1;
               for J in I + 1..X'Last loop
                  X (J) := X (J - 1) + 1;
               end loop;
               return;
            end if;
         end loop;
         raise Constraint_Error;
      end Next;
      procedure Put (X : Combination) is
      begin
         for I in X'Range loop
            Put (Integers'Image (X (I)));
         end loop;
      end Put;
   end Combinations;
 
   type Five is range 0..4;
   package Fives is new Combinations (Five);
   use Fives;
 
   X : Combination (1..3);
begin
   First (X);
   loop
      Put (X); New_Line;
      Next (X);
   end loop;
exception
   when Constraint_Error =>
      null;
end Test_Combinations;