# Fully-projected subsets

## TL;DR

This paper generalizes previous combinatorial results by deriving a formula for the alternating sum of fully-projected subsets in multidimensional boxes, involving placement of balls with specific projection properties.

## Contribution

It introduces a new formula for counting fully-projected subsets, extending earlier work to higher dimensions and more complex projection constraints.

## Key findings

- Derived a formula for the alternating sum of fully-projected subsets.
- Extended previous combinatorial results to multidimensional settings.
- Provides a theoretical framework for counting specific arrangements in combinatorics.

## Abstract

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.