Given two graphs H and F, the maximum possible number of copies of H in an F-free graph on n vertices is denoted by ex(n, H, F). We investigate the function ex(n, H, kF), where kF denotes k vertex disjoint copies of a fixed graph F. Our results include cases when F is a complete graph, cycle or a complete bipartite graph. (C) 2019 Elsevier B.V. All rights reserved.