  A new Berry-Esseen bound for non-linear functionals of non-symmetric and
non-homogeneous infinite Rademacher sequences is established. It is based on a
discrete version of the Malliavin-Stein method and an analysis of the discrete
Ornstein-Uhlenbeck semigroup. The result is applied to sub-graph counts and to
the number of vertices having a prescribed degree in the Erd\H{o}s-Renyi random
graph. A further application deals with a percolation problem on trees.
