Illinois State University Mathematics Department


MAT 305: Combinatorics Topics for K-8 Teachers

Spring 1999
6:00 - 8:50 pm Tuesday STV 332
Dr. Roger Day (day@math.ilstu.edu)



Test #3
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 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 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 parts (a) and (b) of question (1).

a. In the expansion of , what is the value of K for the collected term ?

b. How many solutions are there for the equation if each , i = 1,2,3,4, must be a non-negative integer?

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 parts (a) and (b) of question (2).

a. A teacher intends to assign letter grades to one class of students according to the following distribution: 3 As, 6 Bs, 7 Cs, and 2 Ds. In how many ways can this grade assignment occur?

b. Suppose a recurrence relation s(n) is defined as s(n) = s(n-1) - 2·s(n-3) for n larger than 3 with s(1) = 2, s(2) = 5, and s(3) = 1. Determine the absolute difference between the largest and smallest values in the set {s(1), s(2), . . . , s(8)}.

3.

Here is a portion of Pascal's Triangle.

Consider the sum C(4,0) + C(5,1) + C(6,2) + C(7,3) + C(8,4) = C(9,4). The components of this sum are outlined in the triangle above.

a. In the triangle above, outline the following pattern, including the sum represented here with a question mark. Verify that you have included the correct sum.

C(2,0) + C(3,1) + C(4,2) + C(5,3) + C(6,4) + C(7,5) = ?

What is the sum, expressed as a combination?

b. Determine C(7,0) + C(8,1) + C(9,2) + . . . + C(27,20). Respond using combination notation.

c. Determine C(7,7) + C(8,7) + C(9,7) + . . . + C(27,7). Respond using combination notation.

BONUS! State and prove the general result that has been illustrated in the problem.

4.

Use this difference table for parts (a) and (b).

a. Complete as many open rows of the table (D1, D2, D3, D4) as necessary to determine the type of equation that will model the relationship between the values in the first two lines of the table (x and f(x)). Write a brief explanation describing how you know what type of equation this will be.

b. Use the information in the difference table to generate an explicit formula for the relationship between the values in the first two lines of the table, that is, between x and f(x).

Here are the first few rows for a triangular table of values. Use this for questions (c) and (d).

c. Write in entries for two more rows of the table. These are rows 4 and 5.

d. Use the method of finite differences to determine whether a polynomial exists that models the row sums for this table. If such a polynomial exists, state its degree. If a polynomial cannot be used, state why not. You are NOT required to determine any explicit formula here!

5.

Marco's Uncle Oscar gave him two mice, which he named Whiskers and Oscar. But Marco discovered he'd made a big mistake. Oscar should have been named Oscarella! She just had 8 babies, 4 males and 4 females! "Ten mice aren't so many," said Marco to his mother. "They're cute."

"Yes," said his mother, "but these cute baby mice will breed when they are 6 weeks old, and continue breeding for a long time. Babies are born 3 weeks after breeding. Each mother mouse will have a litter of 8 babies, half males and half females.

Today Marco has 10 mice: Whiskers, Oscarella, and 8 babies.

a. How many mice will Marco have in (i) 3 weeks? (ii) 6 weeks? (iii) 9 weeks? (iv) 18 weeks?

b. Write a recurrence relationship for m(n), the number of mice Marco will have n weeks from today. Note that m(0) = 10 and that in part (a) you were asked to determine m(3), m(6), m(9), and m(18).

6.

Many professional sports conduct tournaments to finish a season. Some sports have a single-elimination playoff concluding with a sudden-death championship, such as the National Football League's Super Bowl. Other sports, however, end the season with a multi-game series between two teams.

For instance, in Major League Baseball, the World Series is a best-of-7 playoff. Two teams compete until one team wins four games. Thus, there may be as few as four games in the playoff or as many as seven. In 1998, the New York Yankees swept the San Diego Padres in four games. From the Yankees perspective, the series record was WWWW, where W represents a Yankee win. In 1997, the Florida Marlins defeated the Cleveland Indians four games to three. Their 7-game series record was WLWLWLW, where W represents a Florida victory and L a Florida loss.

For this type of best-of-7 playoff, how many different won-loss records could there be, from the winner's perspective?

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