### Instruction: Compute
\[\frac{1990^3 - 1000^3 - 990^3}{(1990)(1000)(990)}.\].### Response: This problem looks like it involves some algebraic manipulation and factoring.
I notice that 1990 = 1000 + 990, so I can try to use the identity \[(a + b)^3 = a^3 + b^3 + 3ab(a + b)\] to simplify the numerator.
If I plug in a = 1000 and b = 990, I get \[(1000 + 990)^3 = 1000^3 + 990^3 + 3(1000)(990)(1000 + 990)\] which means that \[1990^3 - 1000^3 - 990^3 = 3(1000)(990)(1990)\] Now the numerator is much simpler and I can cancel out some factors with the denominator.
I get \[\frac{1990^3 - 1000^3 - 990^3}{(1990)(1000)(990)} = \frac{3(1000)(990)(1990)}{(1990)(1000)(990)} = 3\]