Fully-projected subsets

Let $k$ and $i_1,\ldots,i_n$ be natural numbers. Place $k$ balls into a multidimensional box of $i_1\times\cdots \times i_n$ cells, no more than one ball to each cell, such that the projections to each of the coordinate axes have cardinalities $i_1,\ldots,i_n$, respectively. We generalize earlier work of Wang, Lee, and Tan to find a formula for the alternating sum of the number of these fully-projected subsets.