I believe there are seven solutions to this. These are listed below:
Each three numbers inside parentheses correspond to the values of a, b, and c in that order in a specific solution:
(0, 0, 2) (0, 3, 0) (1, 1, 1) (2, 2, 0) (3, 0, 1) (4, 1, 0) (6, 0, 0)