  We develop a simple and unified approach to investigate several aspects of
the cluster statistics of random expansive (multi-)sets. In particular, we
determine the limiting distribution of the size of the smallest and largest
clusters, we establish all moments of the distribution of the number of
clusters, and we prove a local limit theorem for that distribution. Our proofs
combine effectively two simple ingredients: an application of the saddle-point
method through the well-known framework of $H$-admissibility, and an ingenious
idea by Erd\H{o}s and Lehner that utilizes the elementary inclusion/exclusion
principle.
