In looking into combinatorics for last week, I ran across several questions about the topic of “derangements” (permutations of objects in which none of them are in their original positions). Let’s look at those, first at probability, and then at the closely related matter of counting. This will also bring us to the Inclusion-Exclusion Principle. …
Certain kinds of word problems tend to be easy to misinterpret or to misstate. That is particularly true in combinatorics. Let’s look at two of those, one recent and one a few years old, where we are assigning people to groups, and the wording is not quite clear.
We’ll first look at several old questions about proving a relationship between permutations or combinations, where we’ll see some algebraic proofs using formulas, and others that center on the meaning of the symbols as ways of counting. The latter are called “combinatorial proofs”. We’ll end with a recent question of the same type, which suggested …
Combinatorial Proofs: Counting the Same Thing in Two Different Ways Read More »
Last week we examined three probability problems that had problems. Looking further back, I find that Jonathan, who asked the first of those questions, asked a group of questions about rolling multiple dice in 2022. They provide some additional lessons about easy mistakes to make.
It’s been a while since we’ve looked at probability. Here, we’ll look at three questions that we received last year. In each case, we have to detect an error! They’re good examples of what can go wrong, and what to do when your answer appears to be wrong.
A couple recent questions dealt with details in the way permutations and combinations are explained. What do we mean when we say that “order matters” for a permutation, and that there is “no repetition” or that the things being chosen are “different”? Teachers need to know how students hear such words.
Last week we looked at the probability that one of infinitely many possible triangles is an acute triangle, and ended up thinking about continuous probability distributions in general. I thought it might be good to look at some old questions dealing with similar issues in various ways. One will even come back to that same …
Some time ago we looked into the probability that a random set of sides (from, say, a broken stick) form a triangle. A recent question asked about the probability that a random triangle is acute (all angles acute) or obtuse (at least one angle obtuse), which led to more discussion of what it means for …
A recent question about lottery numbers reveals that a seemingly special event is in fact surprisingly common: namely, the presence of consecutive numbers in a lottery drawing. The calculation is an interesting one, and we’ll also see a way to check our answer, then compare it to reality.
Last week’s discussion reminded me of another question, from July, about a probability problem that was solved in a hard (but educational) way and an easy way. This instance is more extreme, and, due to its length, requires extreme editing in order to fit here.