In that problem, the graph is acyclic and there are 600 vertices, so the simple O(N*M) dynamic programming should work (reorder vertices so that for all edges u->v u < v, then for each source initialize the array with zeroes, place 1 at the source and for each edge u -> v in increasing order of u do dp[v] += dp[u]).
In general graphs, the problem "how many simple paths exist between u and v" is #P-complete, according to a proof from the internet. (if the paths don't have to be simple, for a given path length they can be counted using matrix exponentiation)
I think the original problem can also be solved in O(log N) matrix multiplications of size N (by adding an edge from every sink to itself and squaring the matrix ceil(log2(N)) times)