Arranging Letters in Words, Revisited
A recent question illustrates well the different ways to solve problems in combinatorics. We’ll see an easy way, another easy way, and a … less suitable … way to solve a set of problems.
Is There More Than One Standard Form for an Equation?
A couple recent questions asked what constitutes “standard form” for a quadratic equation; that will lead us to some older questions about “standard form” for a linear equation. We’ll see that “standard” isn’t quite as standard as you might think.
A Cubic Challenge
Let’s look at a nice little challenge: to find a cubic function with maximum and minimum at given locations – without using calculus. We’ll explore how to solve it with graphing software, and using algebra in a couple ways, and finally with calculus. And, surprise! They all give the same answer, though the results look …
An Easy But Impossible Probability Problem
I like looking a little deeper into problems; here we’ll find that although the problem is simple if you take it on its own terms, those terms are actually impossible. Does it matter?
Internal and External Division of a Segment
Last time, we considered how to represent algebraically the division of a line segment in a given ratio. At the end, we touched on a subject I recalled discussing extensively almost four years ago: that such a “division” can be either internal (inside the segment, as you’d expect) or external (elsewhere on the line containing …
Dividing a Segment: Get the Order Right!
A series of recent questions dealt with proportional division of a line segment. The context was vectors, and we’ll use them a lot, though the main ideas can be understood using ordinary geometry. We’ll see a mistake so easy to make that AI did it just as humans do; and how textbooks can make it …
How to Fight Errors, and Face the Unknown
Having recently tutored a number of students through their finals, and given a lot of the same advice repeatedly, this seems like a good time to share a couple recent questions on how to approach learning and doing math. We’ll see one student who wants to stop making mistakes, and another who needs to learn …
Derangements: How Often is Everything Wrong?
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. …
Word Problems in Combinatorics: How to Misread (or Miswrite?) Them
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.
Combinatorial Proofs: Counting the Same Thing in Two Different Ways
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 …
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