Interpreting Probability Questions
(A new question of the week) A couple recent questions centered around how to interpret probability problems, whose wording can often be subtle, and whose solutions require care.
Combinatorics
Writing a Proof: Substance, then Style
(A new question of the week) A question from last month provides an opportunity to show how to develop an algebraic proof of a combinatorial identity involving factorials. We’ll be looking over Doctor Rick’s shoulder as he guides a student through the maze. I’ll also add in a previously published version of the same proof …
Combinatorics and Coefficients
(A new question of the week) A question from last August gave us some nice problems reminiscent of the Binomial Theorem, which were very deserving of discussion.
Rank of a Binary Number
(A new question of the week) A few months ago, I wrote about Ranking a Word Among Its Permutations, that is, finding where a word would be found in an ordered list of all possible “words” made by permuting its letters. The problem in general requires a (sometimes lengthy) algorithm. A month or so later, …
Ranking a Word Among Its Permutations
(A new question of the week) There are some topics that appear to be standard in certain parts of the world, but far less familiar in our own. Sometimes it takes two of us to recognize what a student is asking, due to language issues and different past experience with such questions. This is an …
Interpreting and Solving a Counting Problem
(An archive question of the week) Combinatorics can be inherently tricky; making up your own problem is doubly so. Here we have a problem created by a teacher, who then is not entirely sure what it means. How can we figure out what meaning to give it? Combine that with working out how to solve …
Arranging Letters with Duplicates
(A new question of the week) Here is a recent discussion with a frequent user of our service, Kurisada, about combinatorics. He is new to the subject, so this involved several introductions to new ideas.
How Many Different Pizzas?
(An archive question of the week) We’ve been looking at examples of extended discussions with students about various kinds of problems. Here, we have one (not from a student) that led to some good thinking about combinatorics – the techniques of counting the ways something can happen.
Permutations and Combinations: Undercounts and Overcounts
(A new question of the week) We have been looking at some combinatorics questions, both easy and challenging. Some questions have come to us in recent weeks that can illustrate how to think your way through relatively difficult problems, including catching errors and interpreting a textbook’s solutions. We’ll see yet again that there are usually …
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Permutations and Combinations: An Introduction
We have seen a number of questions recently about combinatorics: the study of methods for counting possibilities. These topics are studied at all levels of mathematical education, from elementary (where they might just be called counting) to high school (where they are often learned along with probability) to college (where they are part of “discrete …