How many triangles exist that have a perimeter of 15 units and sides measuring integers?
Let's assume that $a, b$ and $c$ are the lengths of the sides of the triangle. Without loss of generality, we can assume that $a \leq b \leq c$, so that $a+b+c \leq 3 c$. Since each side of a triangle is less than the sum of the other two, we have $c<a+b$ and, therefore, we get $2 c<a+b+c \leq 3 c$. But, $a+b+c=15$, so $2 c<15 \leq 3 c$ and, since $c$ is an integer, 5, 6, or 7 are the only options for $c$.

If $c=5$, then $a+b=10$ and we have one possibility, $a=b=c=5$. If $c=6$, then $a+b=9$ and we have two possibilities for $a$, 3 or 4, in which case, necessarily, $b$ is 6 or 5, respectively. If $c=7$, then $a+b=8$ and we have four possibilities for $a, 1, 2, 3$ or 4. In these cases, necessarily, $b$ is $7, 6, 5$ or 4, respectively. Thus, in total, we have seven such triangles.