The number of triangles with integer sides and with perimeter 15 is:

There are 7 triangles. If the longest side is $c$, the triangle inequality gives $c<15-c$, and since it is the longest side, $c\geq5$. The remaining two sides have one unordered pair for $c=5$, two for $c=6$ and four for $c=7$. The count 6 leaves out the equilateral triangle with side 5.