Illinois State University Mathematics Department


MAT 305: Combinatorics Topics for K-8 Teachers

Spring 2000
6:00 - 8:50 pm Tuesday STV 332
Dr. Roger Day (day@math.ilstu.edu)



Test #3
possible solutions

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.

Respond to each of these questions. While you may show steps leading to your solution, you do not need to generate written explanations for questions (a) through (d) of problem 1.

 

a) Determine the number of derangements of the elements in the set {1,2,3,4,5,6}, where the natural position for 1 is the first position, the natural position for 2 is the second position, and so on.

b) Answer the following questions for the recurrance relationship described by:

a(n)=5a(n-1)&endash;2a(n-3) for n a positive integer,
where a(1) = &endash;5, a(2) = &endash;6, and a(3) = &endash;1.

(i) Determine a(6).

(ii) Determine the smallest value of n such that a(n+1) &endash; a(n) is greater than 5000.

c) In the expansion of , state:

i) the number of uncollected terms
ii) the coefficient K in the collected term .

d) Determine the number of collected terms in the expansion of .

2.

Georgania Higgenbottom served a variety of cool drinks at her spring garden party. Each of the 120 guests selected exactly one beverage, chosen from the following list: ginsing go-getter, spinach surprise, cabbage combo, tomato torment, and cucumber delight.

Georgania's head waiter, Christof, kept a tally sheet of drink selections for the 120 guests.

a) Suppose Christof noticed that each of the beverages had been selected by at least one guest. How many different tally sheets could Christof have generated for the party's drink selection under these circumstances?

b) Suppose instead that when Georgania screamed to Christof, "What are our minimums on drinks?" he replied, "I see that at least 10 people chose the ginsing go-getter, 20 or more went for the spinach surprise, more than 6 people tried the cabbage combo, no one selected tomato torment, and at least 15 drank cucumber delight." Under these conditions, how many different tally sheets could Christof have generated for the party's drink selection?

 

c) Try to picture Amiele, the beverage host, walking around the table asking guests for their beverage choice. How many guests would Amiele have to ask before he would be assured that a beverage choice repeated itself? What assumptions underlie your response?

3.

In the Ancient yet little-known game of gnikoj, scoring occurs several ways. A player can earn 1 point for a kaboom, 1 point for a laboom, and 1 point for a maboom. A player earns 2 points for executing a bifter or for executing a cifter. In traditional gnikoj, there is no way to earn 3 points on a single move, and the only way to earn 4 points is by the rarely seen move called a gemserp. Points are added and the player with the highest point total wins.

a) Show at least three different scoring sequences a gnikoj player could execute and earn 11 points. Note that the order in which points are earned is significant. A kaboom followed by a cifter is a different scoring sequence than a cifter followed by a kaboom.

b) In gnikoj, play continues until an essapmi is reached. Individual player totals could exceed 100 points. Create a recursion relationship T(n), including any required initial conditions, to describe the number of different sequences of gnikoj moves that result in a total of n points. Show justification for your result.

4.

What is wrong with the following problem situation?

A survey of 144 new teachers in a metropolitan school district determined that 16 new teachers had been to Europe, 13 had been to Asia, and 17 had been to Africa. Of those who had been to Europe, 10 had been to Africa. Of those who had been to Asia, 9 had been to Africa. Of all the new teachers surveyed, 5 had been to Africa, Asia, and Europe, and 111 new teachers had been to none of those continents.

Describe specifically and clearly what is wrong and how you determined that.

5.

a) Using an unlimited supply of letters from the set {A,B,C,D}, how many 7-letter sets are possible to create with at least one C in the set?

b) Using an unlimited supply of letters from the set {A,B,C}, how many 5&endash;letter words (meaningless or not) are possible to create with exactly one A in the word?

6.

In a local chess tournament, two players compete until one player wins 4 matches. With no regard for the ability of the players, what fraction of all possible two-player competitions will require exactly 7 matches?


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