I got 50 as well, though I wasn't able to figure out a,b,c independently, other than they weren't all integers.
I did come up with the relations,
3 = ab + ac + bc
-2 = abc
Mostly my technique was to do things like
16 = (a+b+c)(a+b+c), expand the right side, and look for terms that matched the other two equations (e.g. aa+bb+cc=10.)
Enough grinding around with this and I got an answer.