Question: Consider the maximum number of triangles in a graph with n nodes and m edges, not necessarily planar. (A triangle has 3 distinct nodes connected by 3 distinct edges.) One simple bound you can give for this quantity is n choose 3 (n(n-1)(n-2)/6), since each triangle has three nodes as vertices. Give a stronger bound.
Answer: Each edge participates in at most n-2 triangles (the third vertex of the triangle can be any of the vertices that aren't endpoints of the edge) and every triangle has three edges. So there are at most m(n-2)/3 triangles in the graph.