How many 3 number combinations could have sum of 23?

Answer 1

Infinitely many.

Consider any integer n and then the triplet (23, n, -n). Whatever the value of n, the triplet will always sum to 23. And since n is an arbitrary integer there are infinitely many such triplets.

Next you could try (22, n, 1-n). Again, an infinite number of these.

And an infinite number of (21, n, 2-n),

and so on ...