A vertex coloring of a graph G is called distinguishing if no non-identity automorphism of G preserves it. The distinguishing number of G, denoted by D(G), is the smallest number of colors required for such a coloring. We are intent to count number of different distinguishing colorings with a set of k colors for some certain kinds of graphs. An application to distinguishing lexicographic products of graphs is also presented. Coauthors: Bahman Ahmadi, Fatemeh Alinaghipour.