Individual members of an ensemble of identical systems coupled to
a common probe can become entangled with one another, even when they
do not interact directly. We investigate how this type of multipartite entanglement is generated in the context of a system consisting of an ensemble of N two-level atoms resonantly coupled to a single mode of the electromagnetic field. In the case where N=2, the dynamical evolution is studied in terms of the entanglements in the different bipartite divisions of the system, as quantified by the I-tangle. We also propose a generalization of the so-called residual tangle that quantifies the inherent three-body correlations in this tripartite system. This allows us to give a complete characterization of the phenomenon of entanglement sharing in the case of the two-atom Tavis-Cummings model. We also introduce an entanglement monotone which constitutes a lower bound on the I-tangle of an arbitrary bipartite system. This measure is seen to be useful in quantifying the entanglement in various bipartite partitions of the TCM in the case where N > 2, i.e., when there is no known analytic form for the I-tangle.