https://file.aiquickdraw.com/mathbot/file/1725933425458ajhin5cj.jpg

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The images you uploaded contain information about a coffee survey and related questions. To solve the problem accurately, let's break down the information from both images.

### Information Extracted:

1. **Survey Details (First Image):**
   - 221 people like espresso.
   - 127 people like cappuccino and chocolate-flavored coffee.
   - 182 people like cappuccino.
   - 136 people like espresso and chocolate-flavored coffee.
   - 209 people like chocolate-flavored coffee.
   - 78 people like all three types of coffee.
   - 116 people like espresso and cappuccino.
   - 82 people like none of these types of coffee.

2. **Questions (Second Image):**
   - **a.** How many of the customers like only chocolate-flavored coffee?
   - **b.** How many like cappuccino and chocolate-flavored coffee, but not espresso?
   - **c.** How many like espresso and cappuccino, but not chocolate-flavored coffee?
   - **d.** How many like exactly one of the three types of coffee?

### Plan for solving:
We can solve this using set theory and Venn diagrams. There are three sets:
- **E** (espresso lovers)
- **C** (cappuccino lovers)
- **Ch** (chocolate-flavored coffee lovers)

By using the principle of inclusion-exclusion and the information provided, we can determine the number of people who fall into each subset.

Would you like me to proceed with solving this step-by-step, including drawing the Venn diagram to illustrate the breakdown?

A gourmet coffee bar conducted a survey to determine the preferences of its customers. Of the customers surveyed:

- 221 like espresso.
- 127 like cappuccino and chocolate-flavored coffee.
- 182 like cappuccino.
- 136 like espresso and chocolate-flavored coffee.
- 209 like chocolate-flavored coffee.
- 78 like all three types of coffee.
- 116 like espresso and cappuccino.
- 82 like none of these types of coffee.

Questions:

1. How many customers like only chocolate-flavored coffee?
2. How many like cappuccino and chocolate-flavored coffee, but not espresso?
3. How many like espresso and cappuccino but not chocolate-flavored coffee?
4. How many like exactly one of the three types of coffee?

This problem involves analyzing customer preferences for espresso, cappuccino, and chocolate-flavored coffee using set theory and Venn diagrams. With specific data from the survey, learn how to apply the Inclusion-Exclusion Principle to calculate how many customers like each type of coffee or a combination of types. Suitable for high school and introductory college students.

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