Illinois State University Mathematics Department


MAT 409: Topics in Algebra and Combinatorics for K-8 Teachers

Summer 2006
Dr. Roger Day (day@math.ilstu.edu)



Test #1
Possible Solutions

You may use factorial notation as well as combination and permutation notation where appropriate (i.e., there is no need to expand 24!).

You are to work alone on this test. You may not use anyone else's work nor may you refer to any materials as you complete the test. You may ask me questions.

Evaluation Criteria

You may earn up to 10 points on each of questions 1 through 6. For each question:

  • 6 points count toward a correct solution to the problem. I will evaluate the mathematics you use:
    • Is it accurate and appropriate?
    • Have you provided adequate justification?
  • 4 points count toward how you express your solution. I will evaluate how you communicate your results:
    • Is your solution clear and complete?
    • Have you expressed logical connections among components of your solution?

1.

Set D contains the following digits: {0,4,8}; set A contains the following letters: {E,U,X,Z}; set T contains the following icons:

(a) Lymmytra will choose one item from among those contained in the three sets and write a poem about it. How many choices does Lymmytra have as the focus of her poem? (3 points)

(b) Maxwell imagined the 3-symbol codes that could be created using first a digit from D, next a letter from A, and finally an icon from T. How many different 3-symbol codes of this form could he create? (3 points)

(c) Natasha created a line-up of the 9 symbols from sets A and T. Her only requirement was that a vowel must be at the beginning of her line-up or a vowel must be at the end of her line-up, but not a vowel in both those positions. How many different line-ups are possible for Natasha? (4 points)

solution

2.

While waiting to get fingerprinted at Livingston County Jail, I watched employees open a door after entering a code by pushing a digital keypad. The keypad is similar to the one shown here.

From my vantage point, I could see neither the door nor the keypad. I could see people pushing buttons from the side. After carefully watching several people enter the door, I concluded that the sequence of keystrokes they used looked like the figure shown below.

  • I knew the order of the keystrokes, as indicated by "first, second" and so on shown here.
  • I did not know which keypad was struck first.
  • I did know that the fourth push was directly below the first and second, and that the third and fifth pushes were directly below the fourth.

Based on this information, determine the number of different 5-number codes that could possibly unlock this door.

solution

3.

Thirteen people are gathered in a small, musty, windowless room. The lead-based paint is peeling off the walls. Among the thirteen people, the only first names represented are Allen, Betsy, and Carl. The only last names represented are Watson, Xian, Yates, and Zawalski. Each person has exactly one first name and one last name. Show that at least two people have both the same first and last names.

solution

4.

Consider the letters in the word formalist.

(a) How many unique arrangements are there for the letters in this word? (2 points)

(b) How many arrangements exist if each arrangement must begin and end with a vowel? (2 points)

(c) How many arrangements exist if all consonants must be kept together? (3 points)

(d) How many 4-letter sets can be created using only the letters in the word? (3 points)

solution

5.

A school cafeteria contained a line of identical tables. Three boys who dined there were such bitter enemies that they could not be trusted to sit close to each other. No two or three of them could sit at the same table, and, in fact, the cafeteria monitor required that there always be at least one buffer table (occupied by others or not) between any two tables where these bitter enemies sat.

(a) What is the minimum number of tables, all in a line, that is required to meet the cafeteria monitor’s seating restrictions for the three enemies? (2 points)

(b) How many ways could the three enemies be seated, under these restrictions, if there were exactly 8 identical tables in line at the school cafeteria? (4 points)

(c) Generalize the solution to (b) for B enemies and T tables. State any restrictions relating the quantities B and T. (4 points)

solution

6.

Four mediators listen to arguments and independently respond in one of three ways:

Agree, Disagree, Pass

Here are two examples of how the mediators report their responses to an argument:

Example A: 2-1-1                              

  • Mediator Amright:   Agree                    
  • Mediator Beelow:    Agree                    
  • Mediator Capstone:  Disagree                
  • Mediator Doormat:  Pass                      

Example B: 2-1-1

  • Mediator Amright:    Pass
  • Mediator Beelow:     Agree
  • Mediator Capstone:   Disagree
  • Mediator Doormat:   Agree

Notes

  • The aggregate response is the same for each example: 2-1-1 (2 agree, 1 disagree, 1 pass, expressed in that order).
  • The examples show two different ways the mediators independently responded, because at least one mediator in the group responded differently to the two cases.

(a) When only their aggregate response is considered, how many ways can the mediators respond to an argument? (5 points)

(b) When each mediator's independent response is considered, how many ways can the group of mediators respond to an argument? (5 points)

solution


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