Illinois State University Mathematics Department
MAT 305: Combinatorics Topics for K8 Teachers Spring 1999 
Semester Exam 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 10. 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?
The BONUS! question is worth 5 points.
1. 
Respond to each of these questions by placing your solution in the blank. While you may show steps leading to your solution, you do not need to generate written explanations for questions (a) through (e).


2. 
The following conjecture is to be proven true by induction or shown to be false using a counterexample: 1 + 2 + . . . + (n1) + n + (n1) + . . . + 2 + 1 = n^2.


3. 
There are 6 men and 5 women on a committee. A subcommittee of 6 is to be formed. The subcommittee must have no less than two men and two women. In how many ways can such a subcommittee be formed? 

4. 
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 six. How many different dominoes can be made under these restraints? 

5. 
For some positive integer n, how many integers between 0 and 2n inclusive must you pick to be sure that at least one of them is odd? 

6. 
Suppose that Jay Cool and the Gingos are playing at the Starlite Theatre. The theatre has one section of seats, arranged with 70 seats in the first (front) row, 72 seats in the second row, 74 seats in the third row, and so on, for a total of 30 rows. The seats are numbered from left to right, with the first seat in the first row being #1, the first seat in the second row #71, and so on.


7. 
The set of letters {A,A,A,B,B,B,B,C,C} is to be used to create kelement subsets.
Create a generating function that can be used to determine the number of 6&endash;element subsets that will have at least one A and at least two Bs.


8. 
Determine numerical values for x, y, and z that make a collected term in the expansion of . 

9. 


10. 
My nephew Seth noticed that Kellogg's cereals offers a set of 5 cartoon characters in its current cereal selections. One cartoon character is in each specially marked box of cereal and the cartoon characters are equally distributed among the cereal boxes currently coming off the production line. Seth has been around me enough so that he can figure out some probabilities. He knows that the probability of getting a complete set by purchasing less than five boxes is 0. He also explained how to determine the probability he could get a complete set by buying exactly 5 boxes of cereal. He said that there are 5! ways he could buy 5 boxes and get all different characters. He also told me that there were 5^5 different ways to get a set of 5 characters, not necessarily all different. The probability of getting a complete set in the first five purchases is 5!/(5^5). Here's where he needs your help: What's the probability that it takes exactly 6 boxes of cereal to collect a complete set of 5 cartoon characters? 

BONUS! 
Each week two World Professional Tennis Organizations determine who are the #1 players in the world for both womens' and mens' professional tennis.





