  We prove a general theorem to bound the total variation distance between the
distribution of an integer valued random variable of interest and an
appropriate discretized normal distribution. We apply the theorem to 2-runs in
a sequence of i.i.d. Bernoulli random variables, the number of vertices with a
given degree in the Erd\"{o}s-R\'{e}nyi random graph, and the uniform
multinomial occupancy model.
