Permutations And Combinations – Better Explained

This book gives examples of how to understand using permutations and combinations, which are a central part of many probability problems. The focus of this book is on understanding why the permutation and combination equations are what they are, which ends up making them a lot easier to understand, remember, and expand than simply memorizing the equations.

Permutations and combinations is a subject usually makes up a chapter in most statistics text books, but it is a chapter that doesn’t do the subject its proper justice. Most chapters on this subject start and end with memorizing the permutation and combination equations, and miss the deeper understanding of them and also skip over the permutation and combination problems that can’t be solved with those equations directly.

What Kind Of Problems Do Many Other Texts Skip?

The permutation & combinations equations are great. Fairly easy to use, and easy to look up if you forget them and aren’t in an exam. If you get a problem like, “You have 20 boxes but can only fit 15 in your truck, how many different combinations of boxes can you take?” It is straight forward to apply the combination equation of N ! / k! / (n-k)! But how would you solve the problem of “You have 20 boxes, and 4 trucks that can fit 6, 5, 4, 3 boxes, how many different ways can the trucks be loaded? Assume it matters what box goes on which truck, but not the order it is loaded within the truck”

The application of the combination equation to that second problem is not obvious. This book walks through how that problem would be solved, and it turns out to be relatively simple and intuitive.

Feedback From Early Reviewers

Several of the early reviewers expressed an interest in having a longer book, and a wider variety of examples. Consequently in this version I have added examples for how combinations & permutations relate to the lottery, the traveling salesperson problem, the odds of getting a flush in Texas Hold'em, the classic urn problems, as well as the binomial theorem. A big thank you for those suggestions!

What Motivated This Book?

I learned the permutation and combination equations in an early college math class, and have used them over the years and never had reason to revisit them looking for a deeper understanding. However after taking a programming challenge for a large tech company recently, challenging permutation & combination problems frequently appear, and the simple equations simply are not sufficient, a deeper understanding is necessary.

Consequently this book also devotes a large section to an example permutation problem of the kind that you might find in a programming challenge. Those problems are frequently in programming challenges because permutations are an easy way to ensure that naïve brute force solutions can’t solve the problems in a reasonable amount of time, and that a more elegant understanding of the math is required.