Counting

Probability of Consecutive Numbers in a Lottery

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.

Fibonacci Word Problems I: Basic

Here and next week, we’ll look at a collection of word problems we have seen that involve the Fibonacci sequence or its relatives, sometimes on the surface, other times only deep down. The first set (here) are direct representations of Fibonacci, while the second set will be considerably deeper.

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 …

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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.

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.

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.)