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# Discrete Structures

1. The university is checking the records to compare how many male students took a math, English, or history class their first semester vs how many female students did so.
2. The records of 1500 male students are analyzed. 400 of these students took Math; 350 students took English, and 600 students took History. 200 students took Math and English; 100 took Math and History, and 200 took English and History. 50 students took Math, History, and English.
3. Use the formula for “inclusion/exclusion: 3 sets” to calculate the number of male students who took Math, English, or History.
4. How many of the male students did not take any Math, English, or History classes?
5. The records of 1500 female students are analyzed. 410 of these students took Math, 300 took English, and 300 took History. 190 took Math and English, 90 took Math and History; and 140 took English and History. 870 of the students did not take any Math, English, or History courses.
6. Use the formula for “inclusion/exclusion: 3 sets” to calculate the number of female students who took all three course types (Math, History, and English)
7. Draw the Venn diagram and fill in the number of female students in each of the eight disjoint regions of the Venn diagram.

iii. According to the Venn diagram, how many female students took exactly one of the three types of courses (math, English, history)?

1. You must show all of your work by setting up the problem using combinatorics.

There are 30 balloons in the van, 10 of which are Blue, 11 are White, and 9 are Green. You will select 9 balloons from the van. Calculate:

1. How many ways can 2 Blue, 3 White and 4 Green balloons be selected?

Carry out the arithmetic so that you have a single number as your answer:

1. How many ways can at least 7 Blue balloons be selected? c. How many ways can at most 2 White balloons be selected?
2. How many ways can the balloons be selected so there are an equal number of each color?
3. How many ways can the balloons be selected so that they are all one color?
4. Suppose 3 of the 30 balloons are striped. How many ways can 9 balloons be selected so that either no striped balloons are selected or just 1 striped balloon is selected?
5. Suppose each of the 9 balloons that are selected will be given to a particular class. Forexample,balloon1couldbegiventoClassAortoClassBortoClassC,etc. There are 9 classes. How many ways can the 9 balloons be selected from the 30? Carry out the arithmetic so that you have a single number as your answer:
6. You must show all of your work by setting up the problem using combinatorics.

The security department has decided that passwords must now be 8 characters. They are considering two scenarios. Calculate how many unique passwords can be generated with each scenario.

Scenario 1: The allowable characters include 26 upper case letters, 26 lower case letters, 10 digits, and four special characters (!,@,#,&). Repetition is permitted. How many unique passwords can be generated?

Scenario 2. No repetition is permitted. For the first five characters in the password, the allowable characters include 26 upper case letters, 10 digits, and the four special characters. However, the last three characters must be lower case letters. How many unique passwords can be generated?

1. You must show your work by setting up the problem using combinatorics.

A group of students are playing a card game in the Johnson center. The game uses a standard, well-shuffled deck of 52 playing cards such as described in your textbook. A hand of cards consists of TEN of the cards. Find the number of different hands that contain:

1. At least 8 Hearts
2. All clubs
3. At most 2 Kings
4. The hand consists of 4 of a kind, 2 of a different kind, and 4 of a different kind.

Last Updated on February 11, 2019

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