---
tags: mth225, dailyprep
---

# Daily Prep 4.4 -- MTH 225

## Overview

This lesson extends some of what we learned earlier about counting *selections* and *arrangements* of things. We'll learn how to use the **factorial** function to count the number of rearrangements of a list, and about the concept of a **$k$-permutation** which is like a binomial coefficient with ordering taken into account.

## Learning objectives

**Basic Learning Objectives:** *Before* our class meeting, use the Resources listed below to learn all of the following. You should be reasonably fluent with all of these tasks prior to our meeting; we will field questions on these, but they will not be retaught.

- *(Review)* Compute the factorial function $n!$.
- Count the number of rearrangements of items in a list of distinct objects.
- Define the concept of a *$k$-permutation* and explain how it's different from the kind of selection counted by the binomial coefficient.
- Count the number of ways to select $k$ items, in order, from a set of $n$ items.

**Advanced Learning Objectives:** *During and after* our class meeting, we will work on learning the following. Fluency with these is not required prior to class.

- Count the number of rearrangements of items in a list of objects that are *not* all distinct.
- Identify the conditions under which the factorial function or a $k$-permutation would be appropriate tools to use in a combinatorics problem.
- Use rearrangements and $k$-permutations in conjunction with other counting techniques to solve a counting problem.

## Resources for learning

**Video:** Watch this from [the MTH 225 playlist](https://vimeo.com/showcase/8667148) (total running time 7:19):

<iframe src="https://player.vimeo.com/video/626473882?h=92e0a801af" width="640" height="360" frameborder="0" allow="autoplay; fullscreen; picture-in-picture" allowfullscreen></iframe>
<p><a href="https://vimeo.com/626473882">Screencast 4.8: Rearrangements and k-permutation</a> from <a href="https://vimeo.com/user132700952">Robert Talbert</a> on <a href="https://vimeo.com">Vimeo</a>.</p>
**Text:** Read [Section 1.3](http://discrete.openmathbooks.org/dmoi3/sec_counting-combperm.html) in the text.

You are free to search for and use other resources in addition to, or instead of the above, as long as you can work the exercises below.

## Exercises

Once you have watched the videos above, go to this form and complete all the non-optional items on it:

https://docs.google.com/forms/d/e/1FAIpQLSfXIYkolQpKrH3Tl8sWNmuOOcd8e2Ujy7kuf3SToIjV7_qbzg/viewform

## Submission and grading

**Submitting your work:** Your work is submitted when you submit the Google Form. You should receive an email receipt indicating that the work was submitted successfully.

**How this is graded:** The pre-class portion of the Daily Prep is graded either 0 points or 1 point, on the basis of completeness and effort. Wrong answers are not penalized. Earning a "1" requires that you:

- Turn the work in before its deadline;
- Leave no item blank or skipped, even accidentally; and
- Give a good-faith effort at a correct answer on every non-optional item.

More information can be found in the [Specifications for Satisfactory Work in MTH 225](/Cy6P0rGZQzuOM3NwZ3ZuMw) document.

When you arrive for the class meeting, you'll be put into a group of 2-3 to complete a quiz over this material, which will be graded on a 0/1 scale on the basis of correctness.