AEP 4: Counting fun

--- tags: mth225, 225-spr22, aep --- # AEP 4: Counting fun ## Overview and Background Counting problems are awesome! :fire: So let's do some. ## Tasks for this AEP Here are three counting problems. Pick any two, and do them. (Don't turn in work on all three!) ### Problem 1 This has two parts: 1. How many positive integers less than 1000000 have exactly one digit equal to 9? (Once such integer would be 109. But 199 would *not* be an example because there are two 9's, not exactly one.) 2. How many positive integers less than 1000000 have exactly one digit equal to 9, *and* have a sum of digits equal to 22? (An example of such a number is 9553. A non-example would be 9931 because although the digits add to 22, there's two 9's instead of one. Another non-example would be 798 because although there's only one 9, the digits don't add up to 22.) :::warning Make sure to read the **Expectations and Grading Criteria** for allowable and non-allowable ways to solve this problem. ::: ### Problem 2 You're hungry from a long day of doing Discrete Structures, so you head to your favorite donut shop to get a half-dozen (that is, six) donuts. The donut shop sells five different kinds of donuts: glazed, chocolate, blueberry, jelly-filled, and cake. The shop carries an unlimited supply of each of these kinds of donuts. You can get as many of each kind as you want, as long as you get only six of them in all. For example, you can get six jelly donuts; or two jelly donuts, two chocolate, and two glazed; or two cake donuts and one of each of the remaining kinds. How many different selections of six donuts can you make? :::warning Make sure to read the **Expectations and Grading Criteria** for allowable and non-allowable ways to solve this problem. ::: ### Problem 3 One of our original counting problems was finding out the number of ways to assign professors to offices. Let's return to that situation: 1. How many ways are there to put six professors into four identical offices? Assume that you can leave some offices empty, and it's OK to put more than one professor in the same office. (In fact, note that it's impossible in this situation to avoid putting more than one professor in the same office because of the Pigeonhole Principle. 2. How many ways are there to put six professors into four identical offices so that none of the offices is empty? :::warning Make sure to read the **Expectations and Grading Criteria** for allowable and non-allowable ways to solve this problem. ::: ## Expectations and Grading Criteria AEPs are graded using the "EMRN" rubric found in the syllabus. Make sure you review the [Standards for Assessments in MTH 225](/KoT83ezHRYO3DqPyXMMMag) document before you submit any work, so you're fully aware of the expectations for the different marks. In particular: - All work needs to be shown *and* your thought processes clearly expressed in all of the tasks of the assignment. The results also need to be correct. You are not just doing math; you are explaining things to a reader, so a mix of math and English is needed. - All the information needed for an "outsider" to understand your work needs to be self-contained within the work. **The reader should not have to do any work to fill in gaps.** :::danger :fire: **Special requirements for this AEP:** - A solution that's obtained by **brute force** (using a computer, or a lot of patience, to simply list all the possible things you are counting and then manually counting them up) is *not* allowed. However, it's not a bad idea to do this to *check* your work. - Your solution must explain **how you approached the setup of your problem**. For example, if you use the binomial coefficient -- why? If you use dots and dividers -- why? It's not enough just to have a right answer and correct computations. Explain to the reader how you got from the statement of the problem, to the selection of your tools, to the computation and then to the answer. [Review this video](https://vimeo.com/630075618) for an example of how the explanation should be done. Solutions that are either of these will result in a mark of "N", and your work will be returned without further comment. ::: Please note, it is the case with all AEP's that **your grade is primarily based on your explanations and writing, and only secondarily on the precision and correctness of your computations.** Correct computations with insufficient explanation will need to be revised and may receive an "N" grade. A grade of "E" is given if all of the above expectations are met, and the work is of superior quality in terms of the clarity of explanations and work, appearance of the writeup, and precision of the mathematics. ## Submitting your work **AEP submissions must be typewritten and saved as either a PDF or MS Word file. No part of your submission may involve handwriting; work that is submitted that contains handwriting will be graded N and returned without feedback.** This includes electronic handwritten docments, for example using a stylus and a note-taking app. To type up your work, you can use MS Word or Google Docs (both of which have equation editors for mathematical notation) or any other computer-based math typesetting tool. Just make sure you save your work as a Word document or PDF (no `.odt`, `.rtf`, or other file extensions are allowed). When you have your work typed up, double-check it for neatness, correctness, and clarity. Then simply submit your document on Blackboard, in the **AEP** area, in the **AEP 4** assignment. ## Getting Help You **may** ask me (Talbert) for help on this assignment in the form of **specific mathematical or technical questions, or clarifying questions about the instructions**. If I cannot answer a question because it would give too much away, I'll tell you so. However please note: **I will not "look over your work" before you submit it to give you feedback on the overall submission**. I have made the expectations clear, so just follow those directions and submit your best work, and you'll be allowed to revise it if needed. For AEPs, the syllabus policy on collaboration is: >**No collaboration is allowed at all** — with other people, or with print or electronic sources other than your textbook, the video playlist, or your notes. **You can ask technology related questions to anyone at any time**. For example if you need help figuring out how to type up your work, there are no restrictions on that.