Counting

Cutting Up Space Using n Planes

As the capstone of this series on counting, lets look at something that’s a little harder to count by drawing: What is the maximum number of regions into which all of 3-dimensional space can be divided by n planes? We’ll look at two significantly different perspectives. Working up through the dimensions The first question is …

Cutting Up Space Using n Planes Read More »

Polygons and Handshakes

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. Counting diagonals We’ll start with …

Polygons and Handshakes Read More »

How Many Different Meals Are Possible?

(An archive question of the week) While gathering combinatorics questions, there were several that stood out. This one will serve well to summarize the topic, showing multiple methods for counting, and contrasting other kinds of problems. The problem The question, from 2007, relates to an Arby’s promotion: How Many Different Dinners Can Be Ordered? I …

How Many Different Meals Are Possible? Read More »

Stars and Bars: Counting Ways to Distribute Items

We have been looking at ways to count possibilities (combinatorics), including a couple ways to model a problem using blanks to fill in. Today, we’ll consider a special model called Stars and Bars, which can be particularly useful in certain problems, and yields a couple useful formulas. (I only remember the method, not the formulas.) …

Stars and Bars: Counting Ways to Distribute Items Read More »

Permutations and Combinations: Undercounts and Overcounts

(A new question of the week) We have been looking at some combinatorics questions, both easy and challenging. Some questions have come to us in recent weeks that can illustrate how to think your way through relatively difficult problems, including catching errors and interpreting a textbook’s solutions. We’ll see yet again that there are usually …

Permutations and Combinations: Undercounts and Overcounts Read More »

Six Distinguishable People in Four Distinguishable Rooms

(An archive question of the week) Last time we looked at some elementary problems in combinatorics, where we counted the number of ways to choose or arrange elements of a set. Let’s look at a somewhat more complicated problem, which will demonstrate issues that come up in interpreting such a problem and in choosing a …

Six Distinguishable People in Four Distinguishable Rooms Read More »