# Counting on rectangular areas

## TL;DR

This paper proves a theorem about the number of identical columns in a rectangular area and applies it to derive combinatorial identities by counting specific subsets.

## Contribution

It introduces a new theorem for counting identical columns in rectangles and uses it to derive combinatorial identities.

## Key findings

- Theorem for counting identical columns in rectangles.
- Derivation of new combinatorial identities.
- Application of the theorem to finite set subsets.

## Abstract

In the first section of this paper we prove a theorem for the number of columns of a rectangular area that are identical to the given one. In the next section we apply this theorem to derive several combinatorial identities by counting specified subsets of a finite set.