Creative proofs in combinations

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 $\frac{1}{\sqrt{n\pi +\frac{\pi }{2}}}<\frac{\left(\genfrac{}{}{0ex}{}{2n}{n}\right)}{2^{2n}}<\frac{1}{\sqrt{n\pi }}$ 2- A special case of Jonathan Wilde's problem 3- Combination series 4- A feature of powerful numbers.

15 pages