Illinois State University Mathematics Department
MAT 409: Topics in Algebra and Combinatorics for K8 Teachers Summer 2006 
Test #1 Possible Solutions 
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 3symbol 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 3symbol codes of this form could he create? (3 points)
(c) Natasha created a lineup of the 9 symbols from sets A and T. Her only requirement was that a vowel must be at the beginning of her lineup or a vowel must be at the end of her lineup, but not a vowel in both those positions. How many different lineups are possible for Natasha? (4 points)


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

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Thirteen people are gathered in a small, musty, windowless room. The leadbased 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.

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

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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)

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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: 211
Example B: 211
Notes
(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)

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