>Advanced Topics in Algebra
for K-8 Teachers
MAT 305 Spring 1996
Test #2
Test #2 Possible Solutions

Please write your solutions on one side only of each piece of paper you use. You may use factorial notation as well as combination and permutation notation unless instructed otherwise.

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, other than those provided with the test. You may ask me questions.

Evaluation Criteria

You may earn up to 6 points on each of questions 1 through 10. For each question:
---> 4 or 5 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?
---> 1 or 2 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. Express P(14,4) in three different ways:

(a) as the product of four consecutive integers;
(b) using factorial notation; and
(c) in terms of C(14,4).

2. How many distinct arrangements exist for the letters in the word PREFERRER?
3. If we label the rows of Pascal's Triangle starting with n = 0 and the columns starting with k = 0, what is the value of the entry in row 20, column 3:

(a) expressed using combination notation; and
(b) expressed as a base-10 number.

4. Replace w, x, y, and z in C(18,9) = C(w,x) + C(y,z) to illustrate Pascal's Formula, a fundamental relationship that exists in Pascal's Triangle.
5. Juanita is a political satirist. She claims to know enough jokes today so that she could tell a different set of three jokes in her warm-up act, every night of the year, for at least 40 years. What is the minimum number of jokes she must know?

NOTE: The set of jokes {A,B,C} is considered one set of jokes, no matter what order Juanita tells the three jokes.

6. Here is a problem situation and a proposed solution.

Problem: How many different three-digit briefcase-lock arrangements can be created if an arrangement contains three distinct digits with no more than one digit greater than 7?

Solution: Think of filling these three spaces with digits:

_____ _____ _____

As CASE I, suppose the arrangement has no digits greater than 7. We have 8 digits to choose from for the first position, 7 for the second, and 6 for the third. We multiply to get 8*7*6 arrangements with no digits greater than 7.

For CASE II, suppose the arrangement has one digit greater than 7. We have 2 digits to choose from for the digit greater than 7, 8 to choose from for the next digit, and 7 to chose from for the final digit. We multiply to get 2*8*7 arrangements with one digit greater than 7.

The two cases are disjoint, for no arrangement can have no digits greater than 7 and one digit greater than 7 at the same time. Therefore, we add the results of the two cases. There are 8*7*6 + 2*8*7 possible arrangements.

Comment on the proposed solution by verifying or disputing the argument.

7. Consider the expansion of (2a + b)^5.

(a) How many collected terms are there?
(b) Write out the first three collected terms, beginning with the term containing the factor a^5.

8. Consider the expansion of (e + f + g + h)^25.

(a) How many uncollected terms are there?
(b) How many collected terms are there?
(c) What is the coefficient of the collected term that contains the factor ef^6g^10h^8

9. George works 16 blocks west and 6 blocks south of his home. All streets from his home to his workplace are laid out in a rectangular grid, and all of them are available for walking. On his walk to work, George always stops at Buddie's Bakery & Deli, located 4 blocks west and 2 blocks south of his home. On his walk home from work, George always stops at Brink's Bank, located 3 blocks north and 4 blocks east of her workplace. If he walks 22 blocks from home to work and 22 blocks from work to home, how many different round-trip paths are possible for George?
10. A shelf is to contain 16 books. There are 11 red books and 5 green books. If no two green books can be adjacent to each other, and the books are distinguishable only by color, how many ways can the 16 books be shelved?

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