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 #3
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.

Solve each of the following counting problems. While you may show steps leading to your solutions, you do not need to generate written explanations for any parts of this question. (2 points each)

(a) You are ordering a 4-course dinner at a French restaurant. For each course, you have 8 choices. How many different dinners can you order?

(b) Twelve men arrive for a casting call and five are chosen, to play the roles of Victor, Wilber, Xanthor, Yore, and Zeke. In how many ways can such a cast be created from the 12 men who arrived?

(c) From a room containing 17 women, choose a team of 8 women and designate one as a team captain. In how many ways can this be done?

(d) From a pool of 12 girls and 9 boys, how many ways can you create a team of 7 girls and 4 boys?

(e) From a group of 14 people, how many ways can you create a team that consists of either 5 people or 6 people?

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 any parts of this question.

(a) In the expansion of (b+a+r+n+s)^13, what is the value of T in the collected term Tb^4a^2rns^5? (3 points)

(b) How many solutions are there for the equation r+a+c+i+n+g=36 if each variable must be a non-negative integer? (3 points)

(c) A letter carrier has 8 letters, one for each of the 8 residents of an apartment complex. In how many ways can the letter carrier distribute the letters so that no resident receives his or her letter? State your answer as a natural number. (4 points)

solution

3.

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 any parts of this question.

(a) A teacher will assign letter grades to one class of students according to the following distribution: 5 As, 9 Bs, 4 Cs, and 3 Ds. In how many ways can this grade assignment occur? (5 points)

(b) Suppose a recursion relationship p(n) is defined as p(n) = p(n-1) – 3·p(n-2) for n larger than 2 with p(1) = 3 and p(2) = 5. Determine the absolute difference between the largest and smallest values in the set {p(1), p(2), . . . , p(6)}. (5 points)

solution

4.

Members of a 5th-grade class recently identified their collecting interests. Among those enrolled in the class, 14 collect stickers, 14 collect cards, and 11 collect animal figurines. Exactly 7 of the students collect both stickers and cards, 7 of them collect both cards and figurines, and 6 of them collect stickers and figurines. In the class, there are 4 students who collect stickers, cards, and figurines.

If at least three students enrolled in this 5th-grade class does not collect any of the items listed above, what is the smallest number of students enrolled in this class?

solution

5.

(a) Twenty different people are standing in a single line to get half-price tickets to a London show. Donna and Seth are two of the people in line, and there are exactly 8 people between Donna and Seth. In how many distinct ways can such a line-up occur? (5 points)

(b) Generalize your solution to (a) for Donna and Seth being among n different people standing in line for tickets, with exactly d people between Donna and Seth. (5 points)

solution

6.

(a) How many unique positive integers exist that are composed of exactly n digits such that each digit is 3, 5, or 7?

(b) Of these, how many contain all three of the digits 3, 5, and 7 at least once?

solution


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