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@math.ilstu.edu)

 Test #1 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 Pizza Hut is offering customers opportunity to create for themselves a unique music CD when they log on to the Pizza Hut website and use an appropriate code number. The Pizza Hut ad claims customers can make a 6-song CD by selecting from 200 different song titles. (a) How many different sets of 6 songs could be selected by a customer? (2 points) (b) How many unique 6-song arrangements could a customer create? (2 points) Suppose that the 200 songs are listed in the following categories: Pop (44), Rock & Roll (52), Rythem and Blues (18), Jazz (21), Classical (36), and Show Tunes (29), with each song listed in one and only one category. The numbers in parentheses above indicate how many songs are listed in each category. (c) If a customer selects one song from each category, how many sets of 6 songs could be selected by the customer? (3 points) (d) Pat refuses to consider any Rock & Roll titles for her CD. How many 6-song CD arrangements does Pat have to choose from? (3 points) solution 2 Three friends each have a red, a white, a yellow, a blue, and a green T-shirt. (a) If each chooses a shirt to wear, how many unique 3-shirt sets could they be seen wearing? (5 points) (b) If each chooses a shirt to wear, how many ways are there for each of them to all choose different colors? (5 points) solution 3 Consider the letters in the word QUADRILLIONTHS. (a) How many unique arrangements are there for the letters in this word? (2points) (b) How many arrangements exist if the four-letter sequence QUAD must be kept together in the order it now appears? (2 points) (c) Considering only unique letters (no repetition), how many 10-letter subsets could be created from these letters? (3 points) (d) How many arrangements exist if each must begin and end with a vowel? (3 points) solution 4 At the Westminister Kennel Club dog show, held this month in Madison Square Garden in New York City and broadcast on the USA channel, every entrant must be assigned a unique registration number. One suggested strategy for assigning registration numbers requires a code that has three parts to it. The first digit ranges from 1 through 7, corresponding to one of the seven dog groups the entrant is classified in. These are the herding group, the sporting group, the working group, the toy group, the terrier group, the hound group, and the non-sporting group. The next two digits correspond to the breed of the dog within the particular group. An English Cocker Spaniel, for instance, within the sporting group, is breed 14. No group has more than 27 breeds. The final numbers in the registration code provide more information about the particular dog. The first of the final digits is a 0 or a 1, based on the sex of the dog. The second and third digits in this last set represent the age of the dog in years, starting with 00 if less than 9 months old and ranging up to 12, the oldest dog in this year's show being 12 years old. The last two digits in this final portion of the registration code represent the state of residence of the dog. Values 01 through 50 represent the 50 United States. If a dog is from outside the U.S., this number appears as 00. (a) Without considering any other circumstances or restrictions, how many unique registration codes are possible under this scheme? (8 points) (b) What problems in dog identification could occur under this strategy? (2 points) solution 5 In a certain leap year, the 13th of the month was a Friday three different times. What day of the week was 29 February that year? (10 points) solution 6 Roberta claimed that the following equation was always true for positive integers n > 1. (a) Is Roberta correct? (2 points) (b) If Roberta is correct, justify her result for the general case. If Roberta is not correct, provide an example to show she is not correct (called a counter example). (8 points) solution

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