The ABCDEs of Counting

In this essay, we will use the Analysis Tool to investigate counting problems. The analysis tool allows you to input a set of outcomes and count how many outcomes start with or contain a sequence of characters. It also counts the total number of outcomes. Here, our focus will not be on determining how many outcomes there are. Instead, we will focus on using the analysis tool to investigate regularity in the set of outcomes, and how that regularity relates to ways of counting the outcomes.

All of the outcomes we discuss here will be constructed from three-letter combinations of A, B, C, D, E. First, let’s demonstrate how to use the analysis tool.

Problem 1:

The code below will print all the ways to create a three-letter word from the letters A, B, C, D, E, where letters can be repeated. The code is hidden, so you do not need to write it yourself.

After you run the above code, copy the printed list and paste it into the input area in the analysis tool below.

After you paste the outcomes into the analysis tool, use the tool to verify there are 125 outcomes. You can do this by pressing the button for count lines.

A mathematical expression that gives the number of outcomes is \(5\times 5\times 5\). We will look about how regularity in the set of outcomes relates to this expression.

You can verify your answer using the “Count lines with prefix” button.

We can relate this question to the mathematical expression above by pointing out how \(5\times 5\times 5\) corresponds to five groups of five groups of five. The list of outcomes is organized by which letter comes first. All outcomes starting with an A are first, and all outcomes starting with an E are last. If we look only at those outcome that start with A, we notice that they are organized by which letter comes second. Within the outcomes starting with A, all the outcomes with second letter A are listed first, and all the outcomes with second letter E are listed last. Finally, if we specify the first two letters, say AA, we see that the outcomes are listed by the alphabetical order of the third letters. Hence, the outcomes really are organized as five groups of five groups of five.

Problem 2:

The code below will print all the ways to create a three-letter word from A, B, C, D, E if no letter is repeated. Use the analysis tool to answer the following questions.

A mathematical expression for this set of outcomes \(5\times 4\times 3\). Using a similar argument to the above question, answer the following.

Problem 3:

The code below will print all the ways to create a three-letter word from A, B, C, D, E if no letter is repeated and the letters appear in alphabetical order (e.g. D will never come before A).

A typical mathematical expression to count these outcomes is \(\frac{5\times 4\times 3}{3\times 2 \times 1}\). However, we argue that as they are listed here the outcomes are better represented by a different expression. Answer the following questions.

Problem 4:

The code below will print all the ways to create a three-letter word from A, B, C, D, E where the letters appear in alphabetical order (e.g. AAB can appear but BAA cannot).

We can find a mathematical expression for this problem in a similarly to how we found an expression for the previous problem. We will focus on finding smaller expressions for the outcomes that start with each letter.

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