---
tags: mth225, dailyprep
---

# Daily Prep 4.1 -- MTH 225

## Overview

With this lesson we start a new module, Module 4 on *Combinatorics*. That term "combinatorics" is just a fancy way of saying "counting stuff". In this module we are going to learn how to count --- not the way we did it when we were toddlers, but **counting complex arrangements** of things. For example, counting the number of 8-bit strings with exactly three "1" bits; or counting the number of ways to get a full house in a standard poker hand; or the number of ways to give 15 identical lollipops to 6 different children. Computer science, particularly the formal study of algorithms, involves a lot of complex counting and that's what this module is about.

We'll start with the two most basic principles of combinatorics: The *Additive Principle* and the *Multiplicative Principle* which are used to count situations arising from two *disjoint* events, and situations that arise from a sequence of choices (respectively).

## 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.

* State the Additive Principle and use it to count the number of arrangements in an appropriate setting.
* State the Multiplicative Principle and use it to count the number of arrangements in an appropriate setting.

**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.

* Given a counting situation, determine whether the Additive Principle or the Mutliplicative Principle (or both, or neither) and then solve the counting problem.
* Check the solution of a counting problem by explicit enumeration using Python code.

## Resources for learning

**Video:** Watch these from [the MTH 225 playlist](https://vimeo.com/showcase/8667148) (total running time 16:50):

<iframe title="vimeo-player" src="https://player.vimeo.com/video/618309204?h=17b1d061c9" width="640" height="360" frameborder="0" allowfullscreen></iframe>

<iframe src="https://player.vimeo.com/video/618923433?h=b267fa03f6" width="640" height="360" frameborder="0" allow="autoplay; fullscreen; picture-in-picture" allowfullscreen></iframe>

**Text:** Read [Section 1.1](http://discrete.openmathbooks.org/dmoi3/sec_counting-addmult.html) in the text. **Note:** We haven't done much with the text in the course... *until now*. **Please make sure to read the text in addition to or instead of watching the videos.**

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/1FAIpQLScF7rib1o3N5ClogCl9ChT2W6NnS7djDOoC_6EDNFedECP9Nw/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.