A brief note on estimates of binomial coefficients

Through the ages, humankind has struggled to come to terms with its own existence, a thorny issue that will no doubt keep philosophers tossing and turning in their beds for aeons to come. The ‘how’ of our existence, though still fiercely debated in some corners, is now reasonably well-understood, but it is the ‘why’ that continues to plague us. Why are we here? What purpose are we meant to serve? Thus we continue to live our lives, with no end in sight on our quest of self-discovery. However, this document, were it capable of intelligent introspection, would face no such quandary, for it has a very simple reason for being. While presenting a proof in our Extremal Combinatorics course, in an ill-advised attempt to avoid Stirling’s Approximation, I made quite a mess of what should have been a routine calculation. I have thus written this note in atonement for my grievous error in judgement. Within you shall find a brief survey of some useful bounds on binomial coefficients, which hopefully covers what I tried to say in that ill-fated lecture. I claim no responsibility for any errors herein, but if you do detect an error, I would be grateful if you would let me know, and I will issue a patch as soon as I am able.