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 ...
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8
7
positive combinations:14 positive combinations not counting the same one just flipped:7 combinations including negatives: infinate
The sum could be anything, but it would be a positive number if the two addends are positive.
-16