Illinois State University Mathematics Department

 MAT 305: Combinatorics Topics for K-8 Teachers Spring 2001 Dr. Roger Day (day@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 Four boys and five girls, all of differing heights, stood in line for a palm reading. If all the boys stand next to each other in line, how many different linear arrangements exist for the nine people? solution 2 A local fast-food outlet offered a variety of meal combinations. Every meal combination included a sandwich, an order of French fries, and a soft drink. Suppose there are 6 different sandwiches, 3 different sizes for French fries orders, and 8 different soft drinks to choose from. (a) How many meal combo orders must be placed to assure that at least one meal combo is ordered twice? (4 points) The fast-food outlet also offers breakfast items. The breakfast sandwiches include five different bagel sandwiches, seven different biscuit sandwiches, and four different English muffin sandwiches. (b) Suzzie Softknuckle comes to the fast-food outlet for a breakfast sandwich. How many different breakfast sandwiches does she have to choose from? (3 points) The Merchanteer County All-Stars softball team stopped at the fast-food outlet. Each of the 21 team members purchased either a double cheeseburger or a hot ham-and-cheese sandwich. No team member ordered more than one sandwich. (c) Freddie the Fry Cook kept track of the sandwich purchases in the exact order they were made. How many different orderings were possible if k team members purchased a hot ham-and-cheese sandwich? (3 points) solution 3 Five brothers each have the same set of seven hats, distinguished only by color. Each has a white hat, a black hat, an orange hat, a green hat, a yellow hat, a maroon hat, and a red hat. (a) If each brother chooses a hat to wear, how many 5-hat sets could they be seen wearing? (3 points) (b) How many ways are there for each of the brothers to all choose hats of different colors? (3 points) (c) At a recent family reunion, four of the brothers were seen wearing the same color hat while the fifth brother wore a hat of a different color. In how many ways could this have occurred? (4 points) solution 4 4. Consider the letters in the word EXCESSIVENESS. (a) How many unique arrangements are there for the letters in this word? (2 points) (b) How many arrangements exist if the three-letter sequence EXC must be kept together in the order shown? (2 points) (c) How many arrangements exist if each must begin and end with a consonant? (3 points) (d) How many 3-letter sets can be created using only the unique letters in the word? (3 points) solution 5 At Bunion College, a small undergraduate institution in the midwest, every student must select a password to use to enter the college computer network. In creating a password, the following restrictions must all be met. The password must contain only digits or letters of the alphabet. The password must be four or five characters in length. The password must begin with either a B or a C. How many different passwords are possible under these restrictions? solution 6 Francisco approached his combinatorics instructor and showed her the following claims: (i) 5! + 5! = 10! (ii) 2! - 1! = 1! (iii) 5! = 5*4! (a) State whether each claim is true or false. Show arithmetic work to support your response. (3 points) (b) For any of the three claims that are true, write a generalization of Francisco's claim. (3 points) (c) For each generalization you wrote for (b), determine whether it is true or false. If it is false, provide a counterexample to show that; if it is true, justify that the result holds in general. (4 points) solution

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