  We consider the asymptotic behavior of a system of multi-component trapped
bosons, when the total particle number $N$ becomes large. In the dilute regime,
when the interaction potentials have the length scale of order $O(N^{-1})$, we
show that the leading order of the ground state energy is captured correctly by
the Gross-Pitaevskii energy functional and that the many-body ground state
fully condensates on the Gross-Pitaevskii minimizers. In the mean-field regime,
when the interaction length scale is $O(1)$, we are able to verify Bogoliubov's
approximation and obtain the second order expansion of the ground state energy.
While such asymptotic results have several precursors in the literature on
one-component condensates, the adaption to the multi-component setting is
non-trivial in various respects and the analysis will be presented in details.
