  Loeb showed that a natural extension of the usual binomial coefficient to
negative (integer) entries continues to satisfy many of the fundamental
properties. In particular, he gave a uniform binomial theorem as well as a
combinatorial interpretation in terms of choosing subsets of sets with a
negative number of elements. We show that all of this can be extended to the
case of Gaussian binomial coefficients. Moreover, we demonstrate that several
of the well-known arithmetic properties of binomial coefficients also hold in
the case of negative entries. In particular, we show that Lucas' Theorem on
binomial coefficients modulo $p$ not only extends naturally to the case of
negative entries, but even to the Gaussian case.
