A Course on Mathematical Thinking — Binomials

Suresh Kothari, Iowa State University and ENSOFT (kothari@iastate.edu, kothari@ensoftcorp.com)

Preview

Binomial numbers have fascinated humans since antiquity. These numbers appear mysteriously in solving a very broad range of challenging problems. Binomial numbers are the backbone of Newton’s Calculus and the cornerstone of probability theory and statistics. Using binomials Gauss invented the binomial distribution to approximate the bell curve (normal distribution). This module presents the fundamentals of binomial numbers and gives a glimpse their mysterious appearance is applications. Specifically, we discuss:

• Binomial expansion
• Binomial coefficients
• Binomials in power sets
• Binomials in Venn diagrams
• Binomials in probability and statistics
• A glimpse of the big binomial computation problem and its astonishing solution
• Counting and other problems using Binomials

Glimpse of a systematic method for solving counting problems

Newton’s invention of Calculus was inspired by a mysterious binomial formula he found in the first Algebra book in England written by John Wallis. Binomial expansion is so central to Newton’s Calculus that, by his own wish, it is engraved on his tomb. Binomials as the backbone of Newton’s Calculus is a fascinating topic. We cover it in a separate module.

Lecture Notes

A. Binomials and Counting Problems

B. Binomial Identities

C. A Systematic Method for Solving Counting Problems

Experimental AI Generated Audio Notes

We are including AI generated audio notes. These audios are generated using the podcast feature of Google’s NotebookLM. We think you will find these podcasts intriguing. Beware that AI may misinterpret the material or may add erroneous information. Always keep in mind that these audios are supplementary to the lecture notes. This is an experiment, let us know if you find it helpful.

A. Why Newton chose Binomials over Gravity?

B. Binomial Identities and Committee Problems