 survey of 

8585 

college students was taken to determine the musical styles they liked. Of​ those, 

3535 

students listened to​ techno, 

4040 

to hip​ hop, and 

3131 

to rap.​ Also, 

1212 

students listened to techno and​ rap, 

2222 

to techno and hip​ hop, and 

1818 

to hip hop and rap.​ Finally, 

1010 

students listened to all three musical styles. Construct a Venn diagram and determine the cardinality for each region. Use the completed Venn Diagram to answer the following questions.




To solve this problem and construct the Venn diagram, let's break down the information given.

We have three sets representing students who listen to three different musical styles:
1. **T = Techno**: 3535 students
2. **H = Hip hop**: 4040 students
3. **R = Rap**: 3131 students

Additionally, we are provided with the number of students who listen to combinations of these genres:
- **T ∩ R (Techno and Rap)**: 12 students
- **T ∩ H (Techno and Hip hop)**: 22 students
- **H ∩ R (Hip hop and Rap)**: 18 students
- **T ∩ H ∩ R (All three genres)**: 10 students

### Step-by-step approach to fill the Venn Diagram:

We will start by filling in the intersections involving all three genres and work our way outwards.

#### Step 1: T ∩ H ∩ R (All three genres)
- 10 students listen to all three genres. This goes in the central region where all three circles overlap.

#### Step 2: T ∩ R (Techno and Rap, but not Hip hop)
- 12 students listen to both Techno and Rap. However, 10 of these students also listen to Hip hop. Therefore, the number of students who listen to Techno and Rap **but not** Hip hop is:
  \[
  12 - 10 = 2
  \]

#### Step 3: T ∩ H (Techno and Hip hop, but not Rap)
- 22 students listen to both Techno and Hip hop, but 10 of them also listen to Rap. So, the number of students who listen to Techno and Hip hop **but not** Rap is:
  \[
  22 - 10 = 12
  \]

#### Step 4: H ∩ R (Hip hop and Rap, but not Techno)
- 18 students listen to both Hip hop and Rap, but 10 of them also listen to Techno. Therefore, the number of students who listen to Hip hop and Rap **but not** Techno is:
  \[
  18 - 10 = 8
  \]

#### Step 5: T only (Techno but neither Hip hop nor Rap)
- We know 3535 students listen to Techno in total. From this, we subtract those who listen to Techno and at least one other genre:
  \[
  3535 - (12 + 2 + 10) = 3511
  \]
  So, 3511 students listen only to Techno.

#### Step 6: H only (Hip hop but neither Techno nor Rap)
- We know 4040 students listen to Hip hop in total. From this, subtract those who listen to Hip hop and at least one other genre:
  \[
  4040 - (12 + 8 + 10) = 4010
  \]
  So, 4010 students listen only to Hip hop.

#### Step 7: R only (Rap but neither Techno nor Hip hop)
- We know 3131 students listen to Rap in total. From this, subtract those who listen to Rap and at least one other genre:
  \[
  3131 - (2 + 8 + 10) = 3111
  \]
  So, 3111 students listen only to Rap.

### Summary of Regions:
Now, let's summarize the cardinality of each region for the Venn diagram:

- **T only** = 3511
- **H only** = 4010
- **R only** = 3111
- **T ∩ H only** = 12
- **T ∩ R only** = 2
- **H ∩ R only** = 8
- **T ∩ H ∩ R** = 10

These values will form the regions in the Venn diagram.

---

### Questions:
1. How many students listened to only Hip hop?
2. How many students listened to both Techno and Rap, but not Hip hop?
3. How many students listened to at least two genres?
4. How many students did not listen to any of the three genres?
5. How many students listened to exactly two genres?

### Tip:
When constructing a Venn diagram, always start by filling the most specific intersection (all three genres) and work your way outward to avoid double-counting students in overlapping sets.

A survey of 8585 college students was taken to determine the musical styles they liked. Of those, 3535 students listened to techno, 4040 to hip hop, and 3131 to rap. Also, 1212 students listened to techno and rap, 2222 to techno and hip hop, and 1818 to hip hop and rap. Finally, 1010 students listened to all three musical styles. Construct a Venn diagram and determine the cardinality for each region. Use the completed Venn Diagram to answer the following questions.

Explore how to construct a Venn diagram for 8585 students, where 3535 listen to techno, 4040 to hip hop, and 3131 to rap. Using set theory and combinatorics, determine the exact number of students who listen to specific combinations of genres. Understand the intersections and individual sets with a step-by-step breakdown.

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