Creative proofs in combinations

TL;DR

This paper offers four innovative and succinct proofs for various mathematical issues, including inequalities, a problem case, series, and properties of powerful numbers, contributing new insights and methods to combinatorial mathematics.

Contribution

It introduces creative and concise proofs for four distinct combinatorial and number theory problems, enhancing understanding and techniques in the field.

Findings

Proved a new inequality involving binomial coefficients

Solved a special case of Wilde's problem

Analyzed properties of combination series

Abstract

In this article, we present four issues and provide a creative and concise proof for each of them. The four issues are: 1- Inequality $\frac{1}{\sqrt{n\pi+\frac{\pi}{2}}}<\frac{\binom{2n}{n}}{2^{2n}}<\frac{1}{\sqrt{n\pi}}$ 2- A special case of Jonathan Wilde's problem 3- Combination series 4- A feature of powerful numbers.