Order the distinct integer sides as $a\lt b\lt c$ and use $a+b\gt c$

List possibilities by the largest side; the perimeter bound leaves only small values of $c$

Write the distinct integer sides in increasing order. Checking the small possibilities under the triangle inequality and perimeter bound gives $$\begin{aligned} &(2,3,4),\ (2,4,5),\\ &(3,4,5). \end{aligned}$$ For $c\ge6,$ the smallest new scalene candidate satisfying $a+b\gt c$ is $(3,4,6),$ whose perimeter is already $13,$ and larger choices cannot qualify. Thus there are $3$ triangles.

The correct answer is C.