Illinois State University Mathematics Department

 MAT 409: Topics in Algebra and Combinatorics for K-8 Teachers Summer 2006 Dr. Roger Day (day@ilstu.edu)

 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 4-course dinner at a French restaurant. For each course, you have 8 choices. How many different dinners can you order?

 Solution: 8^4

(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?

 Solution: P(12,5)

(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?

 Solution: C(17,8)*C(8,1)

(d) From a pool of 12 girls and 9 boys, how many ways can you create a team of 7 girls and 4 boys?

 Solution: C(12,7)*C(9,4)
(e) From a group of 14 people, how many ways can you create a team that consists of either 5 people or 6 people?
 Solution: C(14,5)+C(14,6)

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)

 Solution: 13!/(4!2!5!)

(b) How many solutions are there for the equation r+a+c+i+n+g=36 if each variable must be a non-negative integer? (3 points)

 Solution: C(41,5)
(c) A letter carrier has 8 letters, one for each of the 8 residents of an apartment complex. In how many ways can the letter carrier distribute the letters so that no resident receives his or her letter? State your answer as a natural number. (4 points)
 Solution: D(8)=14,833

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)

 Solution: 21!/(5!9!4!3!)
(b) Suppose a recursion relationship p(n) is defined as p(n) = p(n-1) – 3·p(n-2) for n larger than 2 with p(1) = 3 and p(2) = 5. Determine the absolute difference between the largest and smallest values in the set {p(1), p(2), . . . , p(6)}. (5 points)
 Solution: 69

4.

Members of a 5th-grade 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 5th-grade class does not collect any of the items listed above, what is the smallest number of students enrolled in this class?
 Solution: 26 Use the Inclusion/Exclusion Principle or create a Venn Diagram to show the various numbers of students in each of the 8 possible collecting-category combinations.

5.

(a) Twenty different people are standing in a single line to get half-price 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 line-up occur? (5 points)

 Solution: 2*11*18! Donna and Seth can exchange places in line (2 possibilities). There are 11 places in the 20-position line-up where Donna and Seth can be located so that 8 people are between them. The remaining 18 people can be placed 18! ways into the remaining 18 places in the line up.
(b) Generalize your solution to (a) for Donna and Seth being among n different people standing in line for tickets, with exactly d people between Donna and Seth. (5 points)
 Solution: 2*(n-d-1)*(n-2)! Generalizing from above: Donna and Seth can exchange places in line (2 possibilities). There are n-(d+2)+1 = n-d-1 places in the 20-position line-up where Donna and Seth can be located so that n people are between them. The remaining n-2 people can be placed (n-2)! ways into the remaining n-2 places in the line up.

6.

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

 Solution: 3^n In each of the n places, there is a choice among the three available digits.
(b) Of these, how many contain all three of the digits 3, 5, and 7 at least once?
 Solution: 3^n - 3*2^n +3Discussion at 1999 semester exam solutions, problem #9.

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