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 #2
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.

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 (e) on this page. (2 points each)

(a) What value J satisfies the equation ?

(b) How many distinct arrangements exist for the letters in the word tattletale?

(c) In the expansion of state:

  • (i) the number of uncollected terms; 
  • (ii) the coefficient M in the collected term

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

(e) Replace j and k in C(20,8) + C(20,9) = C(j,k) to correctly illustrate Pascal's Formula.

solution

2.

Referring to the letters in the word EXPEDIENCIES, solve each of the following problems. Each problem is independent and separate from the others.

 (a) How many unique arrangements are there for the letters in this word? (2 points)

(b) How many unique arrangements exist if each cannot begin nor end with I? (2 points)

(c) How many unique arrangements can be made if no two vowels can be adjacent to each other? (3 points)

(d) If the only distinction we can make is between vowels and consonants, how many arrangements can be made? (3 points)

solution

3.

A ballot on a recent California election included 16 propositions. For each proposition, voters could choose AGREE or DISAGREE to express their preference.

(a) How many different marked ballots could be submitted, given that a choice of AGREE or DISAGREE had to be made for each of the 16 propositions? (3 points)

(b) How many different marked ballots could show 9 propositions marked AGREE and 7 propositions marked DISAGREE, given that a choice of AGREE or DISAGREE had to be made for each of the 16 propositions? (3 points)

(c) If voters could ABSTAIN from choosing a response on any of the propositions, how many different ballots could be submitted by voters? (4 points)

solution

4.

On the floor are a pile of 9 mathematics books and 4 science books, all with different titles. The books are to be placed on one shelf.

(a) If a book's title distinguishes it from other books:

  • (i) How many ways can the 13 books be shelved? (2 points)
  • (ii) If no two science books may be adjacent, how many ways can the 13 books be shelved? (3 points)

(b) If a book's subject (mathematics or science) is its only distinguishing property:

  • (i) How many ways can the 13 books be shelved? (2 points)
  • (ii) If all 4 science books must remain together, how many ways can the 13 books be shelved? (3 points)

solution

5.

Consider the expansion of .

(a) How many uncollected terms are there? (3 points)

(b) How many collected terms are there? (3 points)

(c) What is the coefficient of the collected term that contains the factor ? (4 points)

solution

6.

A domino is a rectangle formed by two congruent squares. Each square contains an orderly pattern of "pips" or dots representing a number from zero through nine. How many different dominoes can be made under these restrictions?

solution


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