  We generalize the well-known zero bias distribution and the $\lambda$-Stein
pair to an approximate zero bias distribution and an approximate
$\lambda,R$-Stein pair, respectively. Berry Esseen type bounds to the normal,
based on approximate zero bias couplings and approximate $\lambda,R$-Stein
pairs, are obtained using Stein's method. The bounds are then applied to
combinatorial central limit theorems where the random permutation has the Ewens
$\mathcal{E}_\theta$ distribution with $\theta>0$ which can be specialized to
the uniform distribution by letting $\theta=1$. The family of the Ewens
distributions appears in the context of population genetics in biology.
