The maximum number of copies of k_r,s in graphs without long cycles or paths

The circumference of a graph is the length of a longest cycle of it. We determine the maximum number of copies of K_r,s, the complete bipartite graph with classes sizes r and s, in a 2-connected graph with circumference less than k. As corollaries of our main result, we determine the maximum number of copies of K_r,s in n-vertex P_k-free and M_k-free graphs for all values of n, where P_k is a path on k vertices and M_k is a matching on k edges.