Probability: Cards vs Dice
(A new question of the week) A couple recent questions involved related subtleties in probability and combinatorics. Both were about apparent conflicts between similar problems involving cards and dice.
Combinatorics
One Team, Two Teams, My Team, Your Team
(A new question of the week) Counting ways to select teams can be simple, or quite complex. Here we’ll look at a few tricky examples.
Combinatorics: Multiple Methods, Subtle Wording
(A new question of the week) With few new questions of general interest available this week, I thought I’d go back a few months to a couple little questions on a topic we haven’t dealt with lately, combinatorics. We’ll have one question each on permutations and combinations, showing some subtlety in both the methods we …
How Many Squares in a Checkerboard?
Last week we looked at ways to count paths along the edges of a rectangular grid. Now we’ll look at a companion problem: counting the number of squares (or rectangles) of all sizes in a square (or rectangular) grid. This, too, is a very common question, and I’ll be picking just a few of many …
How Many Paths from A to B?
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.
Fibonacci, Pascal, and Induction
A couple weeks ago, while looking at word problems involving the Fibonacci sequence, we saw two answers to the same problem, one involving Fibonacci and the other using combinations that formed an interesting pattern in Pascal’s Triangle. I promised a proof of the relationship, and it’s time to do that. And while we’re there, since …
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.
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. Three problems The question came from Arsh: I have some coefficient problems which I am unable to solve. I don’t know if a single concept will work for …
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, …