A popular kind of question in combinatorics is to count the number of paths between two points in a grid (following simple constraints). This can be done by very different methods at different levels. We’ll look at several problems of this type, starting with the simplest.
(A new question of the week) A recent question involved a word problem about fractions, which will fit in nicely with the current series on fractions. We’ll explore several ways to solve a rather tricky fraction word problem, some avoiding fractions as much as possible, some focusing on the meaning of the fractions, and others …
(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.
(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 …
Last week, we looked at two solutions to the problem of finding the probability that you can make a triangle using three pieces of a stick, if we cut it at two independently chosen, random locations. This time, we look another solution to that problem, and a similar solution to the version in which we …
This week we look at questions about how likely it is that you can make a triangle out of three random pieces of a stick. As always in probability, the first issue comes in deciding how the process is to be done (that is, what does it mean to break a stick randomly?); we’ll also …
We’ll spend the next couple weeks looking at various counting problems. This topic, called combinatorics, is often studied along with probability, but many of the topics we’ll see here feel more like geometry problems! Here, we’ll be counting the diagonals of a polygon, and handshakes between people at a party.
News about testing for viruses has reminded me of a couple problems that I linked to some time ago, but never dealt with directly. The question is, given data such as the result of a (fallible) blood or swab test, how sure can we be of the results? The answer is sometimes surprising.
(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.
(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, …