A short proof of a famous combinatorial identity

TL;DR

This paper presents a simple proof of a well-known combinatorial identity using the inclusion-exclusion principle, making the proof more accessible and straightforward.

Contribution

It offers a concise, accessible proof of a famous combinatorial identity, connecting it directly to the inclusion-exclusion principle.

Findings

The identity can be derived from inclusion-exclusion.

The proof simplifies understanding of the combinatorial sum.

The approach highlights the power of inclusion-exclusion in combinatorics.

Abstract

We explain how the identity $$\sum_{i+j=n}\binom{2i}{i}\binom{2j}{j}\;=\;\displaystyle4^n$$ is an easy consequence of the inclusion-exclusion principle.