Consider each pair
1. $(a,a):$ $(a*a) +(a*a) = a+a = b \neq c.$ So, $(a,a)$ is not possible
2. $(a,b):$ $(a*a) + (a*b) = a +b = a \neq c.$ So, $(a,b)$ is not possible
3. $(a,c):$ $(a*a)+(a*c) = a+c = c$
$\quad \quad \quad \quad (b*a) +(c*c) = b + b =b \neq c.$ So, $(a,c)$ is not possible
4. $(b,a):$ $(a*b) +(a*a) = b +a = a \neq c.$ So, $(b,a)$ is not possible
5. $(b,b):$ $(a*b) + (a*b) = b+b =b \neq c.$ So, $(b,b)$ is not possible
6. $(b,c):$ $(a*b) + (a*c) = b + c = c$
$\quad \quad \quad \quad (b*b) + (c*c) = c+b = c.$ So, $(b,c)$ is a solution
7. $(c,a):$ $(a*c) + (a*a) = c+a = a \neq c.$ So, $(c,a)$ is not possible
8.$(c,b):$ $(a*c) + (a*b) = c+b = c$
$\quad \quad \quad \quad (b*c) +(c*b) = a+c = c.$ So, $(c,b)$ is a solution
9. $(c,c):$ $(a*c) +(a*c) =c+c= b \neq c.$ So, $(c,c)$ is not possible
So, no. of possible solutions is $2$.