Number of integer points union of fundamental parallelepiped

Question

Assume you have a convex pointed cone in $\mathbb{R}^m$: $C=A \mathbb{R}^n_+$, where A is a $m \times n$ full rank integer matrix. Given a set $B$ of $m$ linearly independent columns of $A$, I know that the number of integer points in the fundamental parallelepiped generated by the submatrix $A_B$ : $\{y \in \mathbb{Z}^m | y=Ax, 0\le x <1\}$ is equal to the determinant of $A_B$. Now, can we say anything about the number of integer points in the union of all these fundamental parallelepiped when $B$ is any set of $m$ linearly independent columns of $A$?