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
(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
(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
(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 solution is 2