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.
Combinatorics
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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Permutation vs Combination: Clarifying Our Terms
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.
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.
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.