  Maxwell introduced a necessary minimum number of edges in terms of the number
of vertices required for a graph to yield a Euclidean rigid generic framework
in $\mathbb{R}^3$, this count was generalised to $\mathbb{R}^d$, for all $d\geq
1$. In this paper, we give the analogous minimum number of $k$-simplices, for
all $0\leq k\leq d$, required for a pure $d$-dimensional simplicial complex to
yield a volume rigid generic framework in $\mathbb{R}^d$, for all $d\geq 1$. In
order to do so, we prove some basic facts about the volume rigidity matroid and
use exterior algebraic shifting, a recently added tool to the study of volume
rigidity. We later prove a volume rigidity Vertex Removal Lemma and use our
count to strengthen the statement.
