  Let $Y=X_1+\cdots+X_N$ be a sum of a random number of exchangeable random
variables, where the random variable $N$ is independent of the $X_j$, and the
$X_j$ are from the generalized multinomial model introduced by Tallis (1962).
This relaxes the classical assumption that the $X_j$ are independent. We use
zero-biased coupling and its generalizations to give explicit error bounds in
the approximation of $Y$ by a Gaussian random variable in Wasserstein distance
when either the random variables $X_j$ are centred or $N$ has a Poisson
distribution. We further establish an explicit bound for the approximation of
$Y$ by a gamma distribution in stop-loss distance for the special case where
$N$ is Poisson. Finally, we briefly comment on analogous Poisson approximation
results that make use of size-biased couplings. The special case of independent
$X_j$ is given special attention throughout. As well as establishing results
which extend beyond the independent setting, our bounds are shown to be
competitive with known results in the independent case.
