So a = 1, b = 2 and c = 3 will fit this identity, as would any other combination of the 3 values. That is a total of 6 possible solutions. It is 6 because there are 6 possible different arrangements of 1,2 and 3.
As a matter of interest, 6 is one of the set of 'perfect numbers' where the sum of all the factors equals their product. The next one is 28 but that has 5 factors so that would not fit.
