Expected number of independent sets of size $k$ in random graph $G(n,p)$
The probability that a specific set of size $k$ is independent is exactly $(1-p)^{\binom{k}{2}}$ (why?). Linearity of expectation shows that the expected number of independent sets of size $k$ is $\binom{n}{k} (1-p)^{\binom{k}{2}}$ (why?).

If you can't follow this calculation, please follow Denis Pankratov's advice and look up linearity of expectation and indicator random variables.