  Stein's method is used to approximate sums of discrete and locally dependent
random variables by a centered and symmetric Binomial distribution. Under
appropriate smoothness properties of the summands, the same order of accuracy
as in the Berry-Essen Theorem is achieved. The approximation of the total
number of points of a point processes is also considered. The results are
applied to the exceedances of the $r$-scans process and to the Mat\'ern
hardcore point process type I.
