Let x be the length of the 3rd side.
From triangle inequality, we know:
8 + 15 > x
x < 23
8 + x > 15
x > 7
combine the 2 results, we get:
7 < x < 23.
The question now becomes:
Find the number of positive integers that satisfies 7 < x < 23.
We can count the answer, which is 15.
Therefore there are 15 possible lengths for the 3rd side, if it is a +ve int.