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The question involves a survey conducted at a dance studio where 86 students were asked if they had competed in the dance categories: Smooth, Rhythm, or Standard. The results are provided using a Venn diagram that divides the total number of students based on which categories they competed in.

To solve this problem, let's break down the Venn diagram:

- **14** students competed only in Smooth.
- **5** students competed only in Rhythm.
- **10** students competed only in Standard.
- **12** students competed in both Smooth and Rhythm but not Standard.
- **8** students competed in both Smooth and Standard but not Rhythm.
- **15** students competed in both Rhythm and Standard but not Smooth.
- **13** students competed in all three categories (Smooth, Rhythm, and Standard).
- **9** students did not compete in any of the three categories.

### To verify, we sum up all the values:
\[
14 + 5 + 10 + 12 + 8 + 15 + 13 + 9 = 86
\]
This adds up to 86, confirming that the Venn diagram correctly represents the entire survey.

Do you want further details or have any specific questions?

### 5 Follow-up Questions:
1. How many students competed in exactly two categories?
2. How many students competed in at least one dance category?
3. How many students competed in exactly one dance category?
4. How many students competed in Smooth or Rhythm but not Standard?
5. What is the probability that a randomly selected student did not compete in any category?

### Tip:
When analyzing Venn diagrams, always carefully note the intersections to avoid double-counting, especially in problems involving multiple sets.

A survey was conducted at a local ballroom dance studio asking 86 students if they had ever competed in the following dance categories: Smooth, Rhythm, or Standard. The results are provided in the Venn Diagram.

Analyze a Venn diagram problem involving 86 ballroom dancers who competed in Smooth, Rhythm, or Standard categories. Understand how to calculate the total students in each category and intersections between them. Explore the use of set theory and probability concepts.

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