Illinois State University Mathematics Department

 MAT 305: Combinatorics Topics for K-8 Teachers Spring 2000 6:00 - 8:50 pm Tuesday STV 332 Dr. Roger Day (day@ilstu.edu)

 Test #2 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. 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. a) Express P(16,5) using factorial notation. b) How many distinct arrangements exist for the letters in the word reverberator? c) In the expansion of , state: i) the number of uncollected terms, ii) the coefficient J in the collected term . d) Determine the number of collected terms in the expansion of . e) Replace w, x, y, and z in C(12,6) = C(w,x) + C(y,z) to illustrate Pascal's Formula, a fundamental relationship that exists in Pascal's Triangle. solution 2 Thum lives in Grid City, where the streets are laid our in a grid, running east/west and north/south. Thum's house is in the northwest corner of the city and his girlfriend, Bolina, lives in the southeast corner of the city. Thum's house is 8 blocks north and 12 blocks west of Bolina's. The image below shows the entire city. a) How many 20-block paths are there from Thum's to Bolina's, assuming all streets exist and are open to traffic? b) Grid City eventually will build a walking mall in the shaded location shown below, thereby eliminating a one-block length of street. Under these conditions, how many 20-block paths are there from Thum's to Bolina's? solution 3 Referring to the letters in the word ACMAESTHESIA, 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? b) How many unique arrangements exist if two or more vowels cannot be adjacent to one another? c) If the only distinction we can make is between consonants and vowels, how many arrangements can be made? d) If the only distinction we can make is between consonants and vowels, and two or more consonants cannot be adjacent to one another, how many arrangements can be made? solution 4 Louise invests her money in \$200 lots. She has \$3000 to invest and her daughter Gina has suggested five different mutual funds for Louise's investments. a) How many different ways can Louise invest her money if she insists on putting at least \$200 in each of the five funds her daughter recommended and uses only these five funds? b) If Louise restricts her investments to these five funds but may choose to not invest any money in one or more of the funds, how many different ways can Louise invest her money? solution 5 Consider the binomial . a) Determine the number of uncollected terms in the expansion of this binomial. b) Show and describe a typical collected term in the expansion of this binomial. c) If k = x, m = 2y, and t=8, determine N in the collected term Nx^3y^5. solution 6 A bag contains a virtually unlimited supply of red marbles, blue marbles, white marbles, and yellow marbles. Marbles of any one color are indistinguishable from each other. a) Marbles are drawn from the bag without looking until a set of 6 marbles is created. How many different 6-marble sets could be created? b) Marbles are drawn from the bag without looking until a set of 24 marbles is created. Of all the 24-marble sets we could create, how many have at least two marbles of each color? c) Marbles are drawn from the bag without looking until a set of 30 marbles is created. Of all the 30-marble sets we could create, how many have marbles of exactly three colors in them? solution

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