jmc

Let $a=1000$ and $b=990.$ Then $a+b=1990,$ so $$\begin{array}{rl}\frac{1990^3-1000^3-990^3}{(1990)(1000)(990)}& =\frac{(a+b{)}^3-a^3-b^3}{(a+b)ab}\\ & =\frac{a^3+3a^{2}b+3a{b}^2+b^3-a^3-b^3}{ab(a+b)}\\ & =\frac{3a^{2}b+3a{b}^2}{ab(a+b)}\\ & =\frac{3ab(a+b)}{ab(a+b)}\\ & =\boxed{3}.\end{array}$$