Binomial Coefficients in a Row of Pascal's Triangle from Extension of Power of Eleven: Newton's Unfinished Work

TL;DR

This paper extends the concept of powers of 11 to generate any row of Pascal's triangle by inserting zeros, providing a general formula and verifying its accuracy for large rows.

Contribution

It introduces a novel formula to extend powers of 11 for generating Pascal's triangle rows with zeros, enabling calculations for large n.

Findings

Verified the extended method with Pascal's triangle for various rows.

Successfully generated the 51st row of Pascal's triangle.

Matched the extended power method with traditional Pascal's triangle results.

Abstract

The aim of this paper is to find a general formula to generate any row of Pascal's triangle as an extension of the concept of $\left(11\right)^{n}$. In this study, the visualization of each row of Pascal's triangle has been presented by extending the concept of the power of 11 to the power of 101, 1001, 10001, and so on. We briefly discuss how our proposed concept works for any $n$ by inserting an appropriate number of zeros between $1$ and $1$ (eleven), that is the concept of $\left(11\right)^{n}$ has been extended to $\left(1\Theta1\right)^{n}$, where $\Theta$ represents the number of zeros. We have proposed a formula for obtaining the value of $\Theta$. The proposed concept has been verified with Pascal's triangle and matched successfully. Finally, Pascal's triangle for a large n has been presented considering the $51^{\text{st}}$ row as an example.