  This paper gives a criterion for the non-vanishing of the Dirac cohomology of
$\mathcal{L}_S(Z)$, where $\mathcal{L}_S(\cdot)$ is the cohomological induction
functor, while the inducing module $Z$ is irreducible, unitarizable, and in the
good range. As an application, we give a formula counting the number of strings
in the Dirac series. Using this formula, we classify all the irreducible
unitary representations of $E_{6(2)}$ with non-zero Dirac cohomology. Our
calculation continues to support Conjecture 5.7' of Salamanca-Riba and Vogan
[SV]. Moreover, we find more unitary representations for which cancellation
happens between the even part and the odd part of their Dirac cohomology.
