Illinois State University Mathematics Department
MAT 409: Topics in Algebra and Combinatorics for K8 Teachers Summer 2006 
Test #3: Possible Solutions 
1. 
Solve each of the following counting problems. While you may show steps leading to your solutions, you do not need to generate written explanations for any parts of this question. (2 points each) (a) You are ordering a 4course dinner at a French restaurant. For each course, you have 8 choices. How many different dinners can you order?
(b) Twelve men arrive for a casting call and five are chosen, to play the roles of Victor, Wilber, Xanthor, Yore, and Zeke. In how many ways can such a cast be created from the 12 men who arrived?
(c) From a room containing 17 women, choose a team of 8 women and designate one as a team captain. In how many ways can this be done?
(d) From a pool of 12 girls and 9 boys, how many ways can you create a team of 7 girls and 4 boys?


2. 
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 any parts of this question. (a) In the expansion of (b+a+r+n+s)^13, what is the value of T in the collected term Tb^4a^2rns^5? (3 points)
(b) How many solutions are there for the equation r+a+c+i+n+g=36 if each variable must be a nonnegative integer? (3 points)


3. 
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 any parts of this question. (a) A teacher will assign letter grades to one class of students according to the following distribution: 5 As, 9 Bs, 4 Cs, and 3 Ds. In how many ways can this grade assignment occur? (5 points)


4. 
Members of a 5thgrade class recently identified their collecting interests. Among those enrolled in the class, 14 collect stickers, 14 collect cards, and 11 collect animal figurines. Exactly 7 of the students collect both stickers and cards, 7 of them collect both cards and figurines, and 6 of them collect stickers and figurines. In the class, there are 4 students who collect stickers, cards, and figurines. If at least three students enrolled in this 5thgrade class does not collect any of the items listed above, what is the smallest number of students enrolled in this class?


5. 
(a) Twenty different people are standing in a single line to get halfprice tickets to a London show. Donna and Seth are two of the people in line, and there are exactly 8 people between Donna and Seth. In how many distinct ways can such a lineup occur? (5 points)


6. 
(a) How many unique positive integers exist that are composed of exactly n digits such that each digit is 3, 5, or 7?






