MAT 181 (Discrete Mathematics)

This page was updated on: May 10, 2018.

There will be one or more homework assignments for each section of the textbook that we cover. Several homework assignments may be due on the same day although each assignment will be graded separately.  Homework assignments will be weighted equally, regardless of the number of problems. Even though homework for several sections may be collected at the same time, don't wait until we finish all of the sections to begin your homework. To help you keep up with the class, do homework on a regular basis as we complete each section of the textbook.

Homework makes up 24% of your course grade (about the same as the final exam!). Each assignment will be graded on a scale of 0% to 100%. Half of the grade will be based on the number of problems you attempted and half will be based on a subset of problems that will be graded for clarity, completeness and correctness. 

Please pay attention to due dates. Homework is generally due two classes after it is assigned; however, you should do the homework problems as soon as they are assigned so if you have questions you can ask at the next class and still complete the homework before the due date. Late homework will incur a substantial penalty as described in the Homework Assignments handout.  No homework will be accepted more than one class after the original due date.  Complete homework assignments turned in at least 24 hours before the due date will earn a bonus. Individual homework assignments may not be turned in incrementally.

Due dates will not be extended even if you are absent on the due date.  If you cannot attend class on the due date and you do not want your homework to be penalized, contact your instructor to make other arrangements for turning in your homework BEFORE the due date.  Because there may be occasions when you are unable to attend class, the two lowest homework scores will be dropped before computing your average score.

To receive full credit on homework assignments:

• Do ALL of the homework problems.
• Follow all instructions in the textbook and any instructions provided by your instructor (below).
• Write neatly and legibly.  
• • Do your homework in PENCIL so it will be easy to correct mistakes.
  • Work vertically down your page, i.e., start each problem on a new line with the problem number clearly indicated in the left-hand margin.  DO NOT write the answers to more than one problem on the same line.
• When explanations are called for, use grammatically correct, clear, concise and complete English sentences.
• Where computations are required, show your work and circle your answer.

Section 2.1

READING ASSIGNMENT: Read paragraph 1 on the top of p. 23. Continue reading at "Statements" on p. 24 through the end of the section.

Homework 2.1- Part A (Due date Thursday, 01/25/18, 5:00 pm)

NOTE: Homework is always due at the start of class.

TEST YOURSELF:  1 - 2 (Do these problems yourself and check your answers on the top of p. 39.  DO NOT TURN THESE PROBLEMS IN.  If you questions on any of the problems, please ask.)

HOMEWORK ASSIGNMENT: 5acd, 6, 7, 8, 13, 14, 15

• Remember: On ALL homework assignments, if there are problems with answers in the back of the book, please check your answer AFTER you have worked the problem (see note above). 
• When constructing truth tables:
• • Arrange the rows in the same order as in the textbook and in class.  (For example, for a 2-variable truth table, the rows should be written in the following order: TT, TF, FT, FF).
  • Show columns for each simple statement (e.g., p, q, r, etc.) and for each intermediate step.  This should help you get the right answer, and it will make your assignment easier to grade.  For example, if you were asked to write a truth table for  ~p v ~(q ^ r), you should have columns for each of the following:
  • • p, q, and r (one column for each variable)
    • ~p
    • q^r
    • ~(q^r)
    • ~p v ~(q ^ r).  Circle this column because it is your final answer.

Homework 2.1- Part B (Due date Tuesday, 01/30/18, 5:00 pm)

TEST YOURSELF:  3 - 6 (Do these problems yourself and check your answers on the top of p. 39.  DO NOT TURN THESE PROBLEMS IN.  If you have questions on any of the problems, please ask.)

HOMEWORK ASSIGNMENT: 21, 24, 25, 27, 31, 33, 41, 43, 51

• See comments for HW 2.1-Part A.  In addition ...
• If you need help with #33, first practice with #32, which is a very similar problem and has an answer in the back of the book.  (However, do not turn in #32).
• On #51, show logical equivalence using truth tables.  DO NOT use Theorem 2.1.1.

----------------------------------------------------------------------------------------------------------------

Section 2.2

READING ASSIGNMENT: Read entire section.

Homework 2.2- Part A (Due date Thursday, 02/01/18, 5:00 pm)

TEST YOURSELF: 1-7, 10 (Do these problems yourself and check your answers on the top of p. 51.  Do not turn these problems in.  If you have any questions, please ask.

HOMEWORK ASSIGNMENT: 2, 3, 5, 8, 15, 16, 20abdg, 22ad, 23ad

• See comments on HW 2.1-Part A regarding truth tables, e.g., arrange the rows in the "standard order", show columns for each intermediate step.
• On #2 and #3, rewrite each of these statements using "if-then" in ENGLISH, NOT symbolically.  The answer to #3 is in the back of the book.  Use this as a guide for #2 if necessary.
• On #15, explicitly state whether the two statements are logically equivalent or not.  Also show the truth tables you used to determine whether the two statements are logically equivalent or not logically equivalent.
• On #20, in order to correctly write the negation of each conditional statement, it is strongly recommended that you first rewrite the conditional statement as a disjunction, then negate the disjunction. DO NOT negate the statement by appending "It is not the case that ..." to the beginning of the statement.  Remember: the negation of a conditional statement is NOT a conditional statement!

Homework 2.2- Part B (Due date Tuesday, 02/06/18, 5:00 pm)

TEST YOURSELF: 8-9 (Do these problems yourself and check your answers on the top of p. 51.  Do not turn these problems in.  If you have any questions, please ask.

HOMEWORK ASSIGNMENT: 9, 33, 36, 40, 44, 45, 47a, 48a

• If you need help with #33, first practice with #32 which is very similar and has an answer in the back of the book.
• On #36, be sure to EXPLAIN your answer.  You will receive NO CREDIT for simply answering "yes" or "no."

----------------------------------------------------------------------------------------------------------------

Section 2.3 (Due date Thursday, 02/08/18, 5:00 pm)

READING ASSIGNMENT: Read entire section. However, you will not need to memorize the argument forms or their names. We will use truth tables to determine the validity of all arguments.

TEST YOURSELF: None.

HOMEWORK ASSIGNMENT: 7, 11, 22, 27, 28, 32, 33.

• On #27, 28 and 32: Use symbols to write the logical form of each argument, then use truth tables (NOT the rules of inference) to determine whether the argument is valid or invalid.  You do NOT need to name the rule (e.g., modus pollens) represented by the argument.  If you do not use truth tables, the problem WILL NOT BE GRADED.
• On all problems except #33: If you determine the argument is invalid, circle all "critical rows" in the truth table that led to that answer.  Critical rows are those in which all of the premises are true.
• On #33: Do not use Example 2.3.11 or any of the examples discussed in class.  Make up your own example.

---------------------------------------------------------------------------------------------------------------- 

Section 2.4 (Due date Tuesday, 02/13/18, 5:00 pm)

READING ASSIGNMENT: Read entire section. 

TEST YOURSELF: 1, 4.  (Do these problems yourself and check your answers on the top of p. 78. Do not turn these problems in. If you have any questions, please ask).

HOMEWORK ASSIGNMENT: DO THE PROBLEMS IN THE ORDER SHOWN HERE: 11, 7, 12, 8, 16, 17, 18, 25, 29.

• Do the problems in the order listed (i.e., start with #11).
• Whenever you are asked to construct I/O tables, show your work, i.e. show your intermediate results in separate columns.  Showing just the output column is NOT sufficient.
• On problems where you are asked to draw circuit diagrams, use the appropriate symbols for the logical gates and LABEL each gate (i.e., NOT, AND, OR).  Don't forget the little circle on the right side of the NOT gate.
• On #25: Show the I/O table and the Boolean expression that you used to design the circuit.
• On #29: Use I/O tables (NOT Theorem 2.1.1) to show that the two circuits are equivalent.

---------------------------------------------------------------------------------------------------------------- 

Section 2.5 (Due date Tuesday, 02/20/18, 5:00 pm)

READING ASSIGNMENT: Read entire section.  The subsection entitled "Circuits for Computer Addition" (page 82-top of page 84) will only be covered superficially; it is not necessary to spend a lot of time on this section.  Note that there are errors in Figure 2.5.2 (see Errata for corrections).

TEST YOURSELF: 2, 6, 8 (Do these problems yourself and check your answers on the top of p. 95. Do not turn these problems in. If you have any questions, please ask.)

HOMEWORK ASSIGNMENT: 1, 4, 5, 7, 11, 13, 14, 23, 26, 27, 28, 32, 33, 34, 38, 40, 41, 43, 44, 46.

• On #13 and #14: Show your work (i.e., add the numbers vertically and show where you carried from one column to the next).  DO NOT CONVERT ANY OF THE NUMBERS TO DECIMAL.
• On #23 and #26: Show your work (i.e., flip the bits and add 1).
• On #27 and #28: Show your work (i.e., do the work to find the two's complement, and then convert to decimal).
• On #32, 33 and 34: Show your work (i.e., represent each decimal negative number as its two's complement and add manually.  Then convert your sum to a decimal number).

----------------------------------------------------------------------------------------------------------------

Section 3.1 (Due date Thursday, 02/22/18, 5:00 pm)

READING ASSIGNMENT: Read entire section.  Note that the textbook covers the material in this section in a very mathematically precise manner, introducing a lot of new notation.  In class, we will focus on the concepts first, and introduce some of the notation later.  You may find it easier to understand the textbook after we discuss the material in class.

TEST YOURSELF: 2, 3 (Do these problems yourself and check your answers on p. 108. Do not turn these problems in. If you have any questions, please ask.)

HOMEWORK ASSIGNMENT: 1, 2, 4a, 9, 12, 13, 14, 15, 18abcd, 32 

• A bold "R" used as a domain name refers to the set of real numbers.
• A bold "Z" used as a domain name refers to the set of integers.
• "R+" and "Z+" refer to the set of positive real numbers and the set of positive integers, respectively.
• On #32:
• If the given statement is false, you need to provide a counterexample, as instructed in the problem statement.
• If the given statement is true, you only need to say so.  You do not need to use the method of exhaustion to demonstrate that the statement is true.  In fact, you won't be able to use the method of exhaustion because the domain is the set of all real numbers, which is infinite, and it is not possible to exhaust every member of an infinite set.

--------------------------------------------------------------------------------------------------------------

Section 3.2 (Due date Thursday, 03/01/2018, 5:00 pm)

NOTE: Due date was changed from Tuesday, 02/27/2018.

READING ASSIGNMENT: As in Section 3.1, you may find it easier to understand the textbook after we discuss the material in class.  You may skip "Necessary and Sufficient Conditions, Only If" on pages 114-115.

TEST YOURSELF: 1 - 6 (Do these problems yourself and check your answers on p. 117. Do not turn these problems in. If you have any questions, please ask.)

HOMEWORK ASSIGNMENT: 1, 2, 3, 4, 9, 10, 12, 32

• DO NOT negate quantified statements by simply appending "It is not true that ..." to the beginning of the statement.  Furthermore, negating a statement by simply adding the word "not" in the middle of the statement often creates an ambiguous statement and should generally be avoided.  The negation of a universal statement should be written as an existential statement, and the negation of an existential statement should be written as a universal statement.
• On #3: You are asked to write a "formal" negation, i.e., a negation using symbols similar to the given statements.
• On #4: You are asked to write an "informal" negation, i.e., a negation in everyday English similar to the given statements.
• On #12: Be sure to read the instructions in the textbook immediately above #11.  You are asked to determine whether the suggested negation is correct - NOT whether it is true or false.  If the proposed negation is not correct, you are asked to propose a correct negation of the original statement.

---------------------------------------------------------------------------------------------------------------- 

Section 3.3 (Due date Thursday, 03/08/2018, 5:00 pm)

READING ASSIGNMENT: You may skip Example 3.3.7 and the subsections on "Formal Logic Notation" (p. 125) and "Prolog" (p. 127).  You may stop reading after Example 3.3.9.

TEST YOURSELF: 3, 4, 5 (Do these problems yourself and check your answers on p. 131. Do not turn these problems in. If you have any questions, please ask.)

HOMEWORK ASSIGNMENT: 1, 3ac, 9, 10abcd, 12ad, 13ab, 14ab, 16ab, 17ab, 31, 32, 35ab, 36ab, 41bc 

• On #16a and #17a: Translate the original statement into "everyday" English, i.e., the way people normally talk.  For example, don't write out an English statement that begins "For all r in Q there exists ..."
• On #16b and #17b:  You may write the negation using symbols or using "everyday" English.
• On #35b and #36b: Write the negation of the original statement in "everyday" English.  

-----------------------------------------------------------------------------------------------------------------

Section 3.4 (Due date: Thursday, 03/08/2018, 5:00 pm)

READING ASSIGNMENT: We will focus on the subsection titled "Using Diagrams to Test Validity" (pages 136-139).  You do not need to read the rest of Section 3.4.

HOMEWORK ASSIGNMENT: 7, 12, 13, 21, 22, 23, 24, 25

• Draw Euler diagrams on ALL of the homework exercises to determine whether the given argument is valid or invalid, regardless of the instructions in the textbook (i.e., do NOT use statement forms such as Universal Modus Pollens or Universal Modus Tollens to establish validity).  Problems without diagrams will NOT be graded!
• If an argument is invalid, you only need to draw ONE diagram that shows that the conclusion does not necessarily follow from the premises.

-----------------------------------------------------------------------------------------------------------------

Section 1.2 (Due date: Thursday, 03/15/2018, 5:00 pm)

READING ASSIGNMENT: Read the entire section.

TEST YOURSELF: 1 - 7 (Do these problems yourself and check your answers on p. 13.  Do not turn these problems in.  If you have any questions, please ask.

HOMEWORK ASSIGNMENT: 1, 3, 4, 5, 7df, 8 (see note below!), 9abcdefgh, 11ad, 12ab

• On #8: There may be typographical errors on this problem.  A correct version of the problem was distributed in class and may be downloaded by clicking the link below.  Also note that a "yes" or "no" answer will suffice for this problem.
• On #12: Be sure to follow the instructions in the textbook, i.e., in addition to determining the Cartesian product, be sure to determine the number of elements in the Cartesian product.

----------------------------------------------------------------------------------------------------------------Section 6.1 (Due date: Thursday, 03/29/2018, 5:00 pm)

READING ASSIGNMENT:  Pages 336-339 assume a knowledge of formal proofs (Chapter 4), which is not covered in this course.  Therefore, please read the following:

• Pages 340-342
• Page 344 through Example 6.1.14 on page 348.

TEST YOURSELF: 1, 4, 5, 6, 7, 8, 10.  (Do these problems and check your answers on p. 352.  Do not turn these problems in.  If you have any questions, please ask.)

HOMEWORK ASSIGNMENT: 10, 11bcd, 12abce, 13ace, 14, 17, 18abd, 35

• On # 11 and #12: You may write your answers in set-builder notation or interval notation, whichever you prefer.  (NOTE: The answers to #11 are in the back of the book and are written in set-builder notation).  It is strongly suggested that you draw graphs of each of the intervals A, B and C to help you visualize the requested unions, intersections, complements, etc.
• On #17: Make sure your Venn Diagrams are legible. It should be clear which areas of the diagram are meant to be shaded and which are meant to be unshaded.  It is recommended that you use pencil so you can erase (rather than cross out) any mistakes!
• On #18: In addition to answering "yes" or "no," you need to answer "why."

---------------------------------------------------------------------------------------------------------------- 

Section 10.1 (Due date: Tuesday, April 3, 2018, 5:00 pm)

READING ASSIGNMENT: Read the entire section except for the following:

• On p. 629, skip the paragraph in the middle of the page starting with "In Chapter 8 ..."
• Skip Example 10.1.7.
• Skip the bottom of p. 633 (after reading Example 10.1.9, continue with p. 634).
• On p. 636-637, skip the formal proofs and skip Example 10.1.13.
• Skip p. 638 (stop reading after Example 10.1.14).

TEST YOURSELF: 1 - 10, 13 - 15. (Do these questions and check your answers on p. 642.  Do not turn these problems in.  If you have any questions, please ask.)

HOMEWORK ASSIGNMENT: Download the file below.  Note that the homework assignment includes problems from the textbook as well as problems created by your instructor.

CLARIFICATION (added 03/27/2018): The homework assignment sheet (below) asks you to do the last three problems (i.e., the problems that are not in the textbook) on a separate sheet of paper. That means you should NOT try to write up the answers to these problems directly on the homework assignment handout.  However, feel free to write up the last three problems on the same sheet(s) of paper you used for the problems taken from the textbook.

---------------------------------------------------------------------------------------------------------------- 

Section 10.2 (Due date: Tuesday, 04/17/2018, 5:00 pm)

NOTE: As of Thursday, 04/05/2018, you should be able to do ALL of the homework problems except the problem involving the map of Illinois on the homework assignment handout.

READING ASSIGNMENT: Read the entire section except for the following:

• On p. 647, stop reading after the solution to Example 10.2.3 (at the top of the page). Skip to "Euler Circuits" on p. 648.
• On p. 650, skip the proof of Theorem 10.2.3  (we will discuss a simpler version in class).
• Example 10.2.6: We will cover this in class, using a simpler algorithm than the one presented in the textbook.
• On p. 644, stop reading at the paragraph near the bottom of the page that begins "Suppose a Graph G with at least two vertices ..." Continue reading on p. 656.

TEST YOURSELF: 1 (except 1h), 2 - 7. (Do these questions and check your answers on p. 660.  Do not turn these problems in.  If you have any questions, please ask.)

HOMEWORK ASSIGNMENT:  Download file below. Note that the homework assignment includes problems from the textbook as well as problems created by your instructor.

Section 10.5 (Homework due date: Thursday, 04/19/2018, 5:00 pm)

READING ASSIGNMENT: Read the entire section except for the following:

• Examples 10.5.3 and 10.5.4: Skim these examples, though it is not important that you understand them thoroughly.  I will show additional examples in class which will not require any knowledge of grammar or chemistry.
• Skip Example 10.5.7.
• Skip Lemma 10.5.3 (although we may come back to this in Sec. 10.7).
• Skip all the formal proofs.  We will discuss the proofs informally in class.

TEST YOURSELF: 4, 5, 6, 7 (Do these questions and check your answers on p. 694.  Do not turn these problems in.  If you have any questions, please ask.)

HOMEWORK ASSIGNMENT:  Download file below. Note that the homework assignment includes problems from the textbook as well as an extra credit problem created by your instructor.

Section 10.7 (Homework due date: Tuesday, 04/24/2018, 5:00 pm)

READING ASSIGNMENT: In this section, we will discuss spanning trees, minimum spanning trees, and Kruskal's algorithm for constructing a minimum spanning tree.  We will not discuss Prim's algorithm or Dijkstra's algorithm.  Kruskal's algorithm is described on p. 704 of the textbook using pseudocode (English text that looks like a computer program).  This method of describing the algorithm may be unfamiliar to you unless you have experience writing computer programs. Kruskal's algorithm will be presented in class using plain English.  Therefore, unless you are familiar with pseudocode,  I recommend that you defer reading about Kruskal's algorithm until we have finished our discussion of Section 10.7 in class.  More specifically:

• Read the following material before class:
• • Beginning of section (p. 701) through the top half of p. 704.  Stop at "Kruskal's algorithm.  You may skip the proof of Proposition 10.7.1 on p. 702.
• After we discuss Section 10.7 in class, read the following:
• • Read the first paragraph under "Kruskal's Algorithm" on p. 704.
  • Read the pseudocode description of Kruskal's algorithm only if you are familiar with pseudocode.
  • Read p. 705 through the first complete paragraph on p. 706.  Stop before Theorem 10.7.2.

TEST YOURSELF: 1, 3, 4 (Do these questions and check your answers on p. 716.  Do not turn these problems in.  If you have any questions, please ask.)

HOMEWORK ASSIGNMENT: Download file below.  Note that the homework assignment includes one problem that is not in the textbook.

---------------------------------------------------------------------------------------------------------------- 

Section 5.1 (Homework due date: 04/26/2018, 5:00 pm)

READING ASSIGNMENT: We will not cover all of the material in Section 5.1. Read from the beginning of the chapter (p. 227) through Example 5.1.16e on p. 238. You may skip Example 5.1.10 on p. 232. 

TEST YOURSELF: 2, 5, 6, 7. (Do these problems yourself and check your answers on p. 244. Do not turn this problem in. If you have any questions, please ask).

HOMEWORK ASSIGNMENT: 1, 2, 3, 7, 10, 11, 12, 13, 16, 19, 20, 21, 26, 30, 44, 46, 63, 67.

• On #44 and 46: Just write the summation using summation notation. You do not need to CALCULATE the sum.
• On #63: Following these steps will help you with #67.
  • Use the recursive definition of factorial to expand the numerator and denominator.
  • Next, reduce the fraction.
  • After you reduce, calculate the value.

--------------------------------------------------------------------------------------------------------------- 

Section 5.6 (Homework due date: Tuesday, 05/01/2018, 5:00 pm)

Extra Credit problem added to assignment 04/22/2018.

READING ASSIGNMENT:

• When reading the text, you may skip Examples 5.6.4, 5.6.5 and 5.6.9.
• Examples 5.6.7 and 5.6.8 will be discussed when we cover Section 5.7 so you may choose to defer reading these examples until we cover Section 5.7,

TEST YOURSELF: No problems assigned.

HOMEWORK ASSIGNMENT: 1, 2, 4, 6, 9, 10, 11, 22abc, 24.  

• If you need help on #4, try #3 first and check the answer in the back of the book.
• #1, #2, #4, #6, #22bc, #24: SHOW YOUR WORK.  SHOW HOW YOU CALCULATED THE TERMS OF THE SEQUENCE.  Don't just write the answer!

EXTRA CREDIT:  Write up the answer to the "Roofer's Dilemma" problem on page 6 of the Sec. 5.6-5.7 handout.  In other words, write out a recursive definition for the sequence R_n, the number of ways to cover a n 1-foot gaps.  Also explain how you derived the recursive definition.

---------------------------------------------------------------------------------------------------------------- 

Section 5.7 (Homework due date: Tuesday, 05/08/2018, 5:00 pm)

Homework due date extended to allow additional time to study for Exam 3.  Homework turned in on the original due date (Thursday, 05/03/2018) will receive extra credit.

READING ASSIGNMENT:

• We will only cover a portion of Section 5.7.  Read through Example 5.7.4, focusing on Examples 5.7.2, 5.7.3 and 5.7.4.  
• After you read the material in Section 5.7, return to Section 5.6 and read Examples 5.6.7 and 5.6.8.

TEST YOURSELF: 3, 4, 5, 6. (Do these problems yourself and check your answers on p. 316.  Do not turn these problems in.  If you have any questions, please ask.)

HOMEWORK ASSIGNMENT: Download the file below.  Note that the first homework problem comes from Section 5.6, not Section 5.7, and that most of the problems on the assignment do not come from the textbook. There is also an extra-credit problem.  In order to get extra credit, you must complete the other problems.

-----------------------------------------------------------------------------------------------------------------

Chapter 9: We will only have time to cover a few highlights in Sections 9.2 and 9.5.  Reading and homework assignments are described below.

Section 9.2 

READING ASSIGNMENT:

• Start reading at the beginning of the section and read through Example 9.2.5.  You may skip part (b) of Example 9.2.4.
• Continue reading on page 530 "When the Multiplication Rule is Difficult or Impossible to Apply" and continue through Example 9.2.11.

TEST YOURSELF: 1, 2, 3. (Do these problems yourself and check your answers on p. 539.  Do not turn these problems in.  If you have any questions, please ask.)

HOMEWORK ASSIGNMENT: See below.

-----------------------------------------------------------------------------------------------------------------

Section 9.5 

READING ASSIGNMENT:

• Read through Theorem 9.5.2 on p. 577.  Stop at "Some Advice About Counting."
• You may skip Example 9.5.5.
• Example 9.5.6 is optional.  If we cover this material, we will do so in a different (and I think easier) way. 

TEST YOURSELF: 1, 4.  (Do these problems yourself and check your answers on p. 584.  Do not turn these problems in.  If you have any questions, please ask.)

HOMEWORK ASSIGNMENT: See below.

-----------------------------------------------------------------------------------------------------------------

HOMEWORK ASSIGNMENT FOR SECTIONS 9.2 AND 9.5

Homework due date: Thursday, 05/10/2018, 5:00 pm.

Extra credit homework will be accepted up through the Final Exam.

There will be one homework assignment covering both sections.  The assignment is not required.  However, completing the assignment can help your grade in two ways:

• I will drop the three lowest homework assignments, instead of the two lowest.  Therefore, completing this homework assignment will help your homework grade unless it is one of your three lowest scores.  Completing the assignment cannot hurt your grade.

• Completing this assignment will qualify you for up to 5% extra credit on final exam problems taken from Chapter 9.

Homework problems:

* Section 9.2: 6ab, 8, 10ab, 14abcde, 18ab, 33abc, 39ab

* Section 9.5: 6ae, 14a, 19a, 20a

• On all problems, show your work.  For example:
• When using the Multiplication Rule, show the numbers you multiplied, e.g., write (2)(4)(5) = 40, instead of just writing 40.
• Write 3!=6 instead of just writing 6, P(5,3)=60 instead of just writing 60, etc.
• You may use the appropriate calculator functions to find factorials and the number of permutations and combinations (you do not need to calculate these manually), but show the functions you used and the inputs, e.g., write 5 nPr 3 =60, don't just write 60.