  The Mahonian statistic is the number of inversions in a permutation of a
multiset with $a_i$ elements of type $i$, $1\le i\le m$. The counting function
for this statistic is the $q$ analog of the multinomial coefficient
$\binom{a_1+...+a_m}{a_1,... a_m}$, and the probability generating function is
the normalization of the latter. We give two proofs that the distribution is
asymptotically normal. The first is {\it computer-assisted}, based on the
method of moments. The Maple package {\tt MahonianStat}, available from the
webpage of this article, can be used by the reader to perform experiments and
calculations. Our second proof uses characteristic functions. We then take up
the study of a local limit theorem to accompany our central limit theorem. Here
our result is less general, and we must be content with a conjecture about
further work. Our local limit theorem permits us to conclude that the
coeffiecients of the $q$-multinomial are log-concave, provided one stays near
the center (where the largest coefficients reside.)
