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

The number of triangles with integral sides possible with perimeter 'x', if x is odd = $$\left\{\dfrac{\left(x+3^2\right)}{48}\right\}\ $$

The number of triangles with integral sides possible with perimeter 'x', if x is even = $$\left\{\dfrac{\left(x^2\right)}{48}\right\}\ $$

{n}->represents the integer nearest to 'n'

The number of triangles with integer sides and with a perimeter of 15.

15 is an odd number.

The number of triangles with integral sides possible with perimeter 15 = $$\left\{\dfrac{\left(15+3^2\right)}{48}\right\}\ $$

=> $$\dfrac{324}{48}$$ = 6.75 $$\approx\ 7$$