  A collection of $N$-diffusing interacting particles where each particle
belongs to one of $K$ different populations is considered. Evolution equation
for a particle from population $k$ depends on the $K$ empirical measures of
particle states corresponding to the various populations and the form of this
dependence may change from one population to another. In addition, the drift
coefficients in the particle evolution equations may depend on a factor that is
common to all particles and which is described through the solution of a
stochastic differential equation coupled, through the empirical measures, with
the $N$-particle dynamics. We are interested in the asymptotic behavior as
$N\to \infty$. Although the full system is not exchangeable, particles in the
same population have an exchangeable distribution. Using this structure, one
can prove using standard techniques a law of large numbers result and a
propagation of chaos property. In the current work we study fluctuations about
the law of large number limit. For the case where the common factor is absent
the limit is given in terms of a Gaussian field whereas in the presence of a
common factor it is characterized through a mixture of Gaussian distributions.
We also obtain, as a corollary, new fluctuation results for disjoint
sub-families of single type particle systems, i.e. when $K=1$. Finally, we
establish limit theorems for multi-type statistics of such weakly interacting
particles, given in terms of multiple Wiener integrals.
