The binomial expansion of \( \left( \frac{1}{2} + \frac{1}{8} \right)^{1016} \) yields terms of the form:
\[T_r = \binom{1016}{r} \left( \frac{1}{2} \right)^{1016-r} \left( \frac{1}{8} \right)^r\]
Simplification of the general term results in:
\[T_r = \binom{1016}{r} \cdot \frac{1}{2^{1016-r}} \cdot \frac{1}{8^r} = \binom{1016}{r} \cdot \frac{1}{2^{1016 - r + 3r}} = \binom{1016}{r} \cdot \frac{1}{2^{1016 + 2r}}\]
For \( T_r \) to be a rational number, its denominator must be a power of 2. This requires \( 1016 + 2r \) to be an integer.
Specifically, \( 1016 + 2r \) must be an even integer. The total count of rational terms corresponds to the number of values of \( r \) that satisfy this condition.
The calculation shows there are 128 rational terms.
Therefore, the correct answer is (2) 128.
