Illinois State University Mathematics Department
MAT 305: Combinatorics Topics for K8 Teachers Spring 2000 
Semester Exam possible solutions 
Evaluation CriteriaYou may earn up to 10 points on each of questions 1 through 10. For each question:
1. 


2. 


3. 
The following conjecture is to be proven true by induction or shown to be false using a counterexample:


4. 
A passenger jet may fly three routes from New York to Chicago and four routes from Chicago to Los Angeles. For a round trip from New York to Los Angeles and back, determine the number of ways a passenger can travel without repeating the same route on any leg of the round trip. 

5. 
Five unique dice are thrown simultaneously. Determine the portion of all possible throws that results in at least two 5s appearing. 

6. 
A nurse walks from home at 10th and H to her clinic at 16th and M, always walking to higher numbers or to letters further along in the alphabet. On a certain day, police block off 13th street between K and L streets. What portion of all possible paths from the nurse's home to the clinic contain the blockedoff street? 

7. 
A company named GAMES has an advertising display with the letters of its name, "GAMES." Colors are used for each letter, but the colors may be repeated. On one particular day, for example, the colors might be red, green, green, blue, red. The company wishes to use a different color scheme for each of the 365 days in the year 2001. Determine the minimum number of colors that are required for this task. 

8. 
In 7,843 families, all of which have a TV set, a dishwasher, a microwave, and a car, there are six different types of TV sets, five different types of dishwashers, four different types of microwaves, and eight different types of cars. What is the least number of families that have the same type of TV, dishwasher, microwave, and car? 

9. 
Cannon balls are stacked in a compact equilateral triangular pattern. When there are n layers in the stack, there are n balls per side of the triangle on the lowest layer, n1 per side on the next layer, and so on, up to 1 ball on the top. Determine a recursion relationship B(n), including any initial conditions, for the total number of balls in a pile with n layers. 

10. 
Exactly 10 chocolate chips are to be distributed at random into 6 chocolatechip cookies. What is the probability that some cookie has at least 3 chips in it? 

BONUS! 
A survey was conducted of 983 families to determine whether they possessed (1) a cell phone, (2) a microwave, (3) a satellite dish, or (4) a CD player. No family was completely without such items, and 481 families had at least two of these items. At least three items were possessed by 345 families and 264 families possessed all 4 items.





