There are quite a few extensions of the Central Limit Theorem to dependent random variables whose dependence is controlled. This includes the case of a sequence of sums of identically distributed random variables whose dependency graphs have uniformly bounded degrees. "On Normal Approximations of Distributions in Terms of Dependency Graphs" is overkill, but it includes a sort of Berry-Esseen result bounding the error of the normal approximation.
My guess is that the expected answer on the qualifying exam was not a proof of that extension of the CLT, but was instead, "Recall this locally dependent version of the Central Limit Theorem from class. See that the indicator variables are only locally dependent." I'd be happy to be wrong about that, though.