I have this Stats and Math question, please help me. Thanks

4 Random Coloring Consider a graph G(V,E) with m edges and a set of q colors. Our goal is to prove that there exists a vertex-coloring of the graph using these q colors so that at most % edges are monochromatic. An edge is monochromatic if both its vertices are assigned the same color. (a) Suppose we color the graph randomly. That is, for each vertex v e V we choose a color uniformly at random and assign it to v. Prove that the expected number of monochromatic edges is g (b) Conclude that there exists a vertex-coloring of G so that at most % edges are monochromatic. (Hint: Suppose that every coloring of G is such so that at least %+ l edges are monochromatic and reach a iontradiction.)