Illinois State University Mathematics Department


MAT 305: Combinatorics Topics for K-8 Teachers

Spring 2001



Semester Exam
Possible Solutions

Please write your solutions on one side only of each piece of paper you use, and please begin each solution on a new sheet of paper. 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 10. 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?

Each BONUS! question is worth 5 points.

 

1.

Respond to each of these questions by placing your solution in the blank. While you may show steps leading to your solution, you do not need to generate written explanations for questions (a) through (e).

a. Replace a, b, and c in C(a,5) = C(5,b) + C(5,c) to illustrate Pascal's Formula.

b. A relationship s(n) is described recursively as s(n) = 3*s(n-1) - 2n, with s(1) = 4. State numerical values for s(2), s(3), and s(4).

c. A group of 24 employees will be shuttled across town in three vans, including a blue van, a red van, and a white van. How many different ways can the employees shuttle across town if 12 people are in the blue van, 7 are in the red van, and 5 are in the white van?

d. A shelf is to contain 12 books, 8 indistinguishable paperback books and 4 indistinguishable hardback books. If the paperback books must be shelved in pairs (that is, exactly and only two paperback books must be adjacent to each other), in how many ways can the 12 books be arranged on a shelf?

e. State the explicit formula for determining the number of derangements of n objects.

solution

2.

Respond to each of these questions by placing your solution in the blank. While you may show steps leading to your solution, you do not need to generate written explanations for questions (a) through (e).

a. Determine the number of collected terms in the expansion of (2a-7c)^n.

b. Determine the value of the coefficient R in the collected term Ra^5bc^12 resulting from the expansion of (a+b+c)^18.

c. Determine the number of uncollected terms in the expansion of (s+t+a+r)^101.

d. Determine the number of collected terms in the expansion of (b+r+a+i+n+s)^16.

e. Consider Pascal's Triangle, where 1 is the 0th row, 1 1 is the 1st row, and 1 2 1 is the 2nd row.

i. Show the elements in row 5 of Pascal's Triangle.

ii. State the difference between the sum of the elements that appear in row 15 of Pascal's Triangle and the sum of the elements that appear in row 16 of Pascal's Triangle.

solution

3.

After updating my advising records this week, I have found that 54 of my advisees each have accumulated from 54 credit-hours to 100 credit-hours. Explain why at least two of these 54 advisees must have the same number of accumulated credit-hours.

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4.

Gary drives a city taxicab during the summer. A client of his, Patti, always rides home in Gary's taxicab. Her apartment is 14 blocks north and 6 blocks east of the office building where she works. The streets between the office building and Patti's apartment are laid out in a rectangular grid, and all streets are accessible to Gary's taxi. How many different paths are available for Gary to make the 20-block trip from Patti's workplace to her apartment?

solution

5.

Thirty-two (32) distinct fair dice are rolled. How many ways are there for nineteen 5s to appear?

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6.

After administering a new medicine, a collection of 314 lab rats was tested for four diseases.

One-hundred fifty-three (153) of the rats tested positive for asefachia, 179 tested positive for bunkeritis, 148 tested positive for cluenegligencia, and 155 tested positive for dipchillase.

Among the same 314 rats, 85 tested positive for both asefachia and bunkeritis, 71 tested positive for both asefachia and cluenegligencia, 75 tested positive for both asefachia and dipchillase, 85 tested positive for both bunkeritis and cluenegligencia, 90 tested positive for both bunkeritis and dipchillase, and 77 tested positive for both cluenegligencia and dipchillase.

We also know that 38 tested positive for all three of asefachia, bunkeritis, and cluenegligencia, 41 tested positive for all three of asefachia, bunkeritis, and dipchillase, 34 tested positive for all three of asefachia, cluenegligencia, and dipchillase and 47 tested positive for all three of, bunkeritis, cluenegligencia, and dipchillase.

Finally, we know that 17 of the 314 lab rats tested positive for all four of the diseases.

How many of the 314 lab rats tested negative for all four of the diseases?

solution

7.

President John Kennedy was well known for his heroic efforts in World War II. During his career in politics, including service in Congress and as President, he developed a reputation for using war stories in his speeches. In fact, those who knew him best claim he told exactly 4 war stories per speech. They also claim that, in the 5004 speeches in which he told exactly 4 war stories, he never once repeated the same 4 stories in the exact same order.

What is the minimum number of war stories Kennedy must have had at his disposal in order for those claims to be true?

solution

8.

Illinois State University surveyed parents/guardians and new students who took part in the university's "preview" activities. If the "preview" office kept track of whether each response was from a male parent/guardian, a female parent/guardian, a male new student, or a female new student, and the "preview" office received a total of 208 responses, how many different sets of 208 responses were possible, with respect to the gender-parent/student make-up of the respondents?

solution

9.

A strange bequest by a recently deceased mathematician involved the two nephews of the mathematician. The two nephews were to divide the mathematician's sports card collection, with the only requirement being that each nephew get at least one card. If there were k different cards in the collection, with no card appearing more than once, how many different divisions were possible between the two nephews?

solution

10.

An unlimited supply of dominoes is available for building a rectangle that measures exactly 2 units high and n units long. Each domino is a 2-by-1 rectangle.

Write a recursive description for R(n), the number of different ways to build a 2-by-n rectangle with dominoes. Be sure to state any initial conditions of the relationship and to explain the basis for your recursive formula.

solution

BONUS!

Each week two World Professional Tennis Organizations determine who are the #1 players in the world for both womens' and mens' professional tennis.

I. A rare disease is known to affect 0.1% of the population. A test for the disease is 95% accurate. You test positive for the disease. What is the probability you actually have the disease?

II. Each week two World Professional Tennis Organizations determine who are the #1 players in the world for both womens' and mens' professional tennis. Steffi Graf was #1 in womens' tennis without interruption over the period Monday, August 17, 1987, until Sunday, March 10, 1991. This is the greatest number of consecutive weeks that any woman or man has been #1 in the professional rankings. How many consecutive weeks were there in Steffi Graf's record?

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