Illinois State University Mathematics Department

 MAT 305: Combinatorics Topics for K-8 Teachers Spring 2003 Dr. Roger Day (day@ilstu.edu)

 Test #1 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.

Set N contains the following digits: {0,1,2,3,4,5,6,7,8,9}; set L contains the following letters: {A,B,C,D,E,F}; set S contains the following symbols: {<,>,=}

(a) A unique 2-character code is to be created by selecting one digit from set N and one letter from set L. If it matters not whether a digit or a letter is listed first, how many unique 2-character codes can be created? (3 points)

(b) Mathilda drew on paper a line-up of six symbols from S. Her line-up included 2 of < and 4 of >. How many different 6-symbol line-ups containing 2 of < and 4 of > could she create? (3 points)

(c) Jackie was asked to create a subset of set N. She could choose no more than one of each digit, her subset was required to contain at least one digit, and her subset could contain no more than 9 digits. How many different subsets could Jackie create? (4 points)

solution

2.

2) The passenger door that is part of a keyless entry system on a new car has a 5&endash;pad keypad on the driver's door similar to the diagram shown here:

 1/2 3/4 5/6 7/8 9/0

Every car is assigned a keyless entry code as it rolls off the assembly line. If each code is a three-digit number, such as 5-7-8, how many different keypad entries (unique sequences of keypad pushes) are possible? For instance, the code 5-7-8 has this keypad sequence:

 5/6 7/8 7/8

solution

3.

Five pairs of shoes were in a single line in Winnie's closet. Each pair was a solid color and there were five different colors: Black, White, Brown, Red, and Green. Each pair contained a left shoe and a right shoe, each distinguishable from one another.

(a) With no regard to matching pairs of shoes, how many ways exist for the shoes to be lined up in a single line? (3 points)

(b) Suppose that four single shoes are chosen from these five pairs. How many ways exist for these four single shoes to contain only shoes for the left foot? (3 points)

(c) Suppose that three single shoes are chosen from these five pairs. Of all the ways to select three single shoes, what portion of those will include a matched pair of shoes? (4 points)

solution

4.

Consider the letters in the word:

ACANTHOCHEILONEMIASIS

a) How many unique arrangements are there for the letters in this word? (2 points)

b) How many arrangements exist if each arrangement must begin and end with a consonant? (2 points)

c) How many arrangements exist if all vowels must be kept together? (3 points)

d) How many 5-letter sets can be created using only the unique letters in the word? (3 points)

solution

5.

Three people who frequented a local juice bar were such bitter enemies that they could not be trusted to sit on bar stools that were next to each other. The juice-bar proprietor required that there always be at least one bar stool (occupied or not) between any two of these bitter enemies.

(a) What is the minimum number of bar stools, all in a line, that is required to meet the proprietor's seating restrictions for the three enemies? (2 points)

(b) How many ways could the three enemies be seated, under these restrictions, if there were 8 bar stools in a line? (4 points)

(c) Generalize the solution to (b) for N enemies and B bar stools. State any restrictions on the quantities N and B. (4 points)

solution

6.

Oliveras approached her discrete-mathematics instructor and showed him the following claims:

(a) C(n,r)=P(n,r)/r!

(b) P(n,r)=n!/(r-1)!

State whether each claim is always true, sometimes true, or never true. Show appropriate evidence to support your response. In addition, if a claim is sometimes true, show a case for which the claim is true and a case for which the claim is false. If a claim is never true, include a case for which the claim is false.(5 points each)

solution

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