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Show all your work to get full credit. Find each of the following. 1. List all the subsets of the given set. 2. A survey of 80 college students was taken to determine the musical styles they listened to. Forty-two students listened to rock, 34 to classical, and 27 to jazz. Twelve students listened to rock and jazz, 14 to rock and classical, and 10 to classical and jazz. Seven students listened to all three musical styles. Of those surveyed, a. How many listened to only rock music? b. How many listened to classical and jazz, but not rock? c. How many listened to classical or jazz, but not rock? d. How many listened to music in exactly one of the musical styles? e. How many listened to music in at least two of the musical styles? f. How many did not listen to any of the musical styles? 3. There are four blood types, A, B, AB, and O. Blood can also be Rh+ and Rh-. Finally, a blood donor can be classified as either male or female. How many different ways can a donor have his or her blood labeled? (Use tree diagram) 4. In a club there are 7 women and 5 men. A committee of 3 women and 2 men is to be chosen. How many different possibilities are there?

Paper For Above Instructions

This paper provides comprehensive answers to the quiz questions outlined in the assignment, with all calculations and methodologies clearly demonstrated.

1. List All the Subsets of the Given Set

Let the set be defined as follows: A = {1, 2, 3}. The subsets of a set include all combinations of elements, including the empty set and the set itself. The number of subsets can be calculated as 2^n, where n is the number of elements in the set.

For the set A = {1, 2, 3}, we have n = 3. Therefore, the number of subsets is:

2^3 = 8

The subsets are:

  • {}

  • {1}

  • {2}

  • {3}

  • {1, 2}

  • {1, 3}

  • {2, 3}

  • {1, 2, 3}

2. Musical Styles Survey Analysis

Using the data provided, we can create a Venn diagram to analyze the number of students listening to different musical styles. Let:

  • R = Rock

  • C = Classical

  • J = Jazz

Given the survey details:

  • n(R) = 42

  • n(C) = 34

  • n(J) = 27

  • n(R ∩ C) = 14

  • n(R ∩ J) = 12

  • n(C ∩ J) = 10

  • n(R ∩ C ∩ J) = 7

  • Total students surveyed = 80

a. Only Rock Music

To find the number of students who listened to only rock music (R only), we use:

R only = n(R) - n(R ∩ C) - n(R ∩ J) + n(R ∩ C ∩ J)

R only = 42 - 14 - 12 + 7 = 23

b. Classical and Jazz, but not Rock

Classical and Jazz, but not Rock is given by:

C ∩ J - n(R ∩ C ∩ J) = 10 - 7 = 3

c. Classical or Jazz, but not Rock

To find this, we need to calculate:

(n(C) - n(R ∩ C)) + (n(J) - n(R ∩ J)) - (C ∩ J - n(R ∩ C ∩ J))

Classical or Jazz not including Rock = (34 - 14) + (27 - 12) - 3 = 32 - 3 = 29

d. Music in Exactly One of the Musical Styles

This is the sum of all students listening to only one style:

Only Rock + Only Classical + Only Jazz = R only + (C - R ∩ C) + (J - R ∩ J) = 23 + (34 - 14 - 10 + 7) + (27 - 12 - 10 + 7)

Calculating gives us:

23 + 17 + 12 = 52

e. Music in At Least Two of the Musical Styles

This can be calculated as follows:

n(at least two) = n(R ∩ C) + n(R ∩ J) + n(C ∩ J) - 2*n(R ∩ C ∩ J)

n(at least two) = 14 + 12 + 10 - 2*7 = 28 - 14 = 14

f. Did Not Listen to Any of the Musical Styles

This is calculated by: Total students - (Rock + Classical + Jazz - those counted multiple times)

Students not listening = 80 - (42 + 34 + 27 - 14 - 12 - 10 + 7) = 80 - 60 = 20

3. Blood Type Labels

We have four types of blood: A, B, AB, O. Each can be either Rh+ or Rh- and classified by gender (male or female). The calculation is as follows:

Blood Types: 4 (A, B, AB, O)

Rh Factors: 2 (Rh+, Rh-)

Gender: 2 (male, female)

Thus, total combinations = 4 2 2 = 16.

4. Combinations of Committee Members

In a club with 7 women and 5 men, we need to choose 3 women and 2 men:

Using the combination formula C(n, k) = n! / (k!(n-k)!):

Number of ways to choose women = C(7, 3) = 35

Number of ways to choose men = C(5, 2) = 10

Total committee combinations = 35 * 10 = 350.

Conclusion

This paper has provided a detailed examination of the quiz questions presented, applying mathematical concepts and logical reasoning to arrive at the correct answers. All steps were included to ensure clarity and full credit as required.

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