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 #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 A bus filled with a high school music group stopped at Blaise's Bistro. The director, a budding mathematician, noticed the menu board at the Bistro and quickly assured her assistant that at least one of the 8 sodas listed on the board would be ordered at least 4 times by the student musicians. Naturally, the director assumed that each student would order exactly one soda from the list. What is the minimum number of students that must be in this music group? solution 2 North American radio stations must adhere to specific guidelines when selecting the call letters for the station name. The name must contain either three or four letters of the alphabet. The name must begin with a W or a K. How many different radio station names are possible under these restrictions? solution 3 Consider the letters in the word SIMULATE. (a) How many rearrangements are there of these letters? (b) How many rearrangements exist if the three-letter sequence SIM must be kept together as it appears? (c) How many rearrangements exist if each must begin and end with a vowel? (d) From the 8-letter set, how many 5-letter subsets exist? solution 4 Using the set {A,B,C}, what fraction of all 5-letter words that can be created contain exactly one A? solution 5 Al and Bobbie are in a group of 12 students. Three teams of 4 students are to be created from the group of 12. Among all possible three-team sets, how many ways exist for Al and Bobbie to be on different teams? solution 6 Complete ONE of the two problems listed below. I. Show an algebraic proof that C(a,c)*C(a-c,b-c) = C(a,b)*C(b,c) for a>=b>=c. II. Determine the smallest natural number n that assures solution

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