For a polynomial-time method of counting integer lattice points for the n-simplex:

Review article - Crites, A., Goff, M., Korson, M., Patrolia, L., Wolcott, L. "Counting Lattice Points in Polyhedra." 

Available here with references for Barvinok's 1994 & 1999 algorithms - <http://www.math.washington.edu/~thomas/teaching/m583_s2008_web/Barvinok.pdf>

For an implementation of Barvinok's algorithm, see J.A. De Loera's LattE program (hosted at UC Davis):
<http://www.math.ucdavis.edu/~latte/group.htm>