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.