---
tags: mth225, dailyprep
---

# Daily Prep 4.5 -- MTH 225

## Overview

In this lesson, we introduce the final one of our combinatorics techniques, used in problems where we are *distributing a number of identical objects to several different recipients or locations*. The **stars and bars** method is really just an application of the binomial coefficient. You'll learn how to turn one of these distribution problems into a diagram, and then into a bitstring, and use the binomial coefficient to solve the problem.

## 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.

- Identify key differences between distribution problems and the other problems we have studied so far.
- Identify incorrect approaches to solving distribution counting problems.
- Given a specific distribution of items to a group, draw its stars-and-bars diagram.
- Given a stars-and-bars diagram, state the specific distribution it represents.

**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.

- Apply the stars-and-bars method to solve distribution counting problems.

## Resources for learning

**Video:** Watch this from [the MTH 225 playlist](https://vimeo.com/showcase/8667148) (total running time 9:53):

<iframe src="https://player.vimeo.com/video/626749580" width="640" height="564" frameborder="0" allow="autoplay; fullscreen" allowfullscreen></iframe>

**Text:** Read through [Section 1.5 in the text](http://discrete.openmathbooks.org/dmoi3/sec_stars-and-bars.html).

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/1FAIpQLSfleIef8dqJnDiBXWPPMkB3pwWWTsBTt_HpcOLSq_CcUkBSvQ/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.