Illinois State University Mathematics Department


MAT 305: Combinatorics Topics for K-8 Teachers

Spring 2003
Dr. Roger Day (day@math.ilstu.edu)



Test #2
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.

Respond to each question below 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) on this page.

a) How many distinct arrangements exist for the letters in the word TRANSFERRERS? (2 points)

b) Consider the expansion of (k + j + m + n)^13.

(i) State the number of uncollected terms. (1 point)

(ii) State the value of the coefficient C in the collected term Ck^2j^3m^5n^3. (1 point)

c) Use Pascal's Formula to express C(21,8) &endash; C(20,7) as a single combination. (2 points)

d) Suppose that S(k) represents the sum of the elements in the kth row of Pascal's Triangle. For instance, S(0) = 1 and S(1) = 2. Express S(9) + S(10) as a multiple of S(9). (2 points)

e) For the equation A+B+C+D+E+F=4, how many possible solutions exist if the variables can take on values that are non-negative integers? (2 points)

solutions

2.

Jack needed to arrange 12 hats on a display shelf. Seven of the hats were red and the other five were blue. The hats were distinguishable only by color.

a) If it was required that none of the blue hats could be adjacent to one another, how many unique arrangements were there of hats on the shelf? (5 points)

b) Instead of (a), suppose it was required that all the blue hats must be kept together. How many unique arrangements of hats on the shelf are there under this restriction? (5 points)

solution

3.

My in-laws live in a retirement community. Among all individual residents within the community, we know that::

  • 38 play golf
  • 21 play tennis
  • 56 play bridge
  • 8 play golf and tennis
  • 17 play golf and bridge
  • 13 play tennis and bridge
  • 5 play golf, tennis, and bridge
  • 72 do not play golf, tennis, nor bridge

a) How many individual residents are there in this retirement community? (5 points)

b) How many of the individual residents participate in only and exactly one of the three activities? (5 points)

solution

4.

Consider the letters in the word INSURRECTIONISTS.

a) How many unique arrangements are there for the letters in this word? (3 points)

b) How many unique arrangements exist if each arrangement must begin and end with the letter R? (3 points)

c) How many unique arrangements exist if all the letters I must be kept together and they may not appear at either end of the word? (4 points)

solution

5.

Iamso Piceune has a particular love for the chocolate-chip cookies served at Blaise's Bistro. Every weekday, Monday through Friday, Iamso has coffee at the Bistro and may or may not eat one or more chocolate-chip cookies.

a) Iamso decided he would consume exactly 15 chocolate-chip cookies per week at the Bistro, and would always eat at least one each day. Determine the maximum number of weeks Iamso could do this without repeating the same 5-day cookie-eating pattern. (5 points)

b) For a long time, Iamso followed the cookie-consumption pattern described in (a). Later, upon advice from his doctor, he changed his cookie-eating pattern. He decided to eat only and exactly 5 cookies per week, with two conditions:

  • (i) It was okay, on one or more days of the week, to eat no cookies, and
  • (ii) He could never eat all 5 cookies in one day.

How many different weekly cookie-eating patterns could Iamso follow under these conditions?

solution

6.

At a local opera, patrons can check their hats prior to entering the auditorium.

a) Suppose that 6 people each check a hat prior to a performance. Those same 6 people are now in line to retrieve the hats. If the 6 hats are returned at random to the 6 people, with no attention paid to whether a hat is returned to its rightful owner, what portion of all possible ways to return the hats will result in no one receiving the correct hat? Here's another way to pose this question: What is the probability that no one receives a correct hat? (5 points)

b) Generalize the problem above to answer the question for n people. What value does the probability approach as n grows larger and larger? (5 points)

solution


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