Illinois State University Mathematics Department

MAT 305: Combinatorics Topics for K-8 Teachers

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### Quiz #01 possible solutions

In the Braille system, a symbol, such as a lowercase letter, a punctuation mark, a suffix, and so on, is represented by raising at least one of the dots in the six-dot arrangement shown in figure (a), where the six Braille positions are labeled as well. For example, in figure (b), the dots in positions 1, 3, and 4 are raised and this six-dot arrangement represents the letter m. The definite article the is shown in (c), and the semicolon (;) is given by the six-dot arrangement in (d), where the dots in positions 2 and 3 are raised.

1. How many different symbols can be represented in the Braille system as described here?

2. How many symbols have an even number of raised dots?

3. How many symbols have no more than one raised dot in each row, where dots 1 and 4 are in the same row, dots 2 and 5 are in the same row, and dots 3 and 6 are in the same row?

A group of five judges listens to arguments and independently responds in one of three ways:

Agree, Disagree, No Judgment

The judges report their responses to an argument as an aggregate response such as 5&endash;0-0 (5 agree, 0 disagree, 0 no judgment) and 1-2-2 (1 agree, 2 disagree, 2 no judgment). The aggregate response is always in the order agree, disagree, no judgment.

4. Based on the individual judges' responses, how many ways are there for an aggregate response to come out as 2-3-0?

5. Based on the individual judges' responses, how many ways are there for an aggregate response to come out as a-b-0, where a and b represent all possible configurations for this context.

6. Sammy's sock drawer contains 12 identical black socks and 12 identical white socks. In the middle of the night, with no lights on, Sammy reaches into the drawer in search of two pairs of the same color socks. How many socks must Sammy pull from the drawer in order to be sure he has two pairs of the same color socks?

Evaluation Criteria

• Impact on Course Grade: Approximately 3%
• Total Points: 30 points
• There are 5 points available for each question. Approximately 3 points of each question is for the accuracy of your solution and 2 points on each question are for the clarity and completeness of your explanation.
• In communicating your solutions, please describe any conditions or assumptions you made as you explored and solved the problems.

### Administered 10 February 2003 Possible Solutions

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