  We consider a variant of the ABC Conjecture, attempting to count the number
of solutions to $A+B+C=0$, in relatively prime integers $A,B,C$ each of
absolute value less than $N$ with $r(A)<|A|^a, r(B)<|B|^b, r(C)<|C|^c.$ The ABC
Conjecture is equivalent to the statement that for $a+b+c<1$, the number of
solutions is bounded independently of $N$. If $a+b+c \geq 1$, it is conjectured
that the number of solutions is asymptotically $N^{a+b+c-1 \pm \epsilon}.$ We
prove this conjecture as long as $a+b+c \geq 2.$
