Combinations

Number of possible combinations from a set.

Code:

Choose:
   ; RDX in range [0,RCX]
      mov r9,rcx
  sub r9,rdx
  jc .err

 push 0 1
    pop rax r8

      ; reduce, R9 = MIN(RDX,R9)
  cmp rdx,r9
  cmovc r9,rdx
        
    jmp .1

.0:   inc r8
      mul rcx
     dec rcx

 ; RDX in range [0,R8)
       cmp rdx,r8
  jnc .err

        div r8

.1:   cmp r8,r9
   jc .0
       ; RAX = RCX choose RDX
      retn

.err:   xor eax,eax
 stc
 retn

...only good for a small range of integers: all the values up to 68 choose 30 are within 64-bit range. Larger values work, for example, 1973 choose 6 doesn't overflow.

This uses a method to generate individual rows of Pascal's triangle.

Code:

  push 52 5
   pop rdx rcx
 call Choose
 jc .error

...an example of use: there are 2598960 possible 5 card hands from a deck of 52 unique cards.