Chris (they/them) is planning a meal for a party. The meal will include an appetizer, a main course, and dessert. They have three choices for an appetizer (Artichoke, Bruschetta, or Caviar), two choices for a main course (Drumsticks or Eggplant), and three choices for dessert (Fritter, Gelato, or Hot chocolate).
b.How many different meals are possible?
c.How many different meals are possible if Chris cannot serve Eggplant and Fritter together? Explain your answer.
d.How many different meals are possible if Chris must serve at least one of Artichoke, Eggplant, and Hot chocolate?

Let's break this problem down systematically:

**Part b: How many different meals are possible?**

To calculate the total number of different meals, we need to consider all the choices Chris has for each course:

- **Appetizer:** 3 choices (Artichoke, Bruschetta, Caviar)
- **Main course:** 2 choices (Drumsticks, Eggplant)
- **Dessert:** 3 choices (Fritter, Gelato, Hot chocolate)

To find the total number of meal combinations, multiply the number of choices for each course:

\[
3 \text{ (appetizers)} \times 2 \text{ (main courses)} \times 3 \text{ (desserts)} = 18 \text{ total meals}
\]

So, Chris has **18 different possible meals**.

---

**Part c: How many different meals are possible if Chris cannot serve Eggplant and Fritter together?**

First, we need to calculate how many meals include both Eggplant (main course) and Fritter (dessert):

- **Appetizer:** 3 choices (Artichoke, Bruschetta, Caviar)
- **Main course:** 1 choice (Eggplant)
- **Dessert:** 1 choice (Fritter)

Thus, the number of meals that contain both Eggplant and Fritter is:

\[
3 \text{ (appetizers)} \times 1 \text{ (Eggplant)} \times 1 \text{ (Fritter)} = 3 \text{ meals}
\]

Since Chris cannot serve Eggplant and Fritter together, we must subtract these 3 invalid meals from the total number of meals we found in part b:

\[
18 \text{ total meals} - 3 \text{ (meals with Eggplant and Fritter)} = 15 \text{ valid meals}
\]

So, **15 different meals are possible** when Eggplant and Fritter cannot be served together.

---

**Part d: How many different meals are possible if Chris must serve at least one of Artichoke, Eggplant, or Hot chocolate?**

First, let's calculate the number of meals where none of these three items are served (i.e., no Artichoke, no Eggplant, no Hot chocolate). In this case:

- **Appetizer:** 2 choices (Bruschetta, Caviar)
- **Main course:** 1 choice (Drumsticks)
- **Dessert:** 2 choices (Fritter, Gelato)

So, the number of meals that exclude Artichoke, Eggplant, and Hot chocolate is:

\[
2 \text{ (appetizers)} \times 1 \text{ (main course)} \times 2 \text{ (desserts)} = 4 \text{ meals}
\]

To find the number of meals that include **at least one** of Artichoke, Eggplant, or Hot chocolate, we subtract these 4 meals from the total number of meals (from part b):

\[
18 \text{ total meals} - 4 \text{ (meals with none of these three)} = 14 \text{ meals}
\]

Therefore, **14 different meals are possible** where Chris serves at least one of Artichoke, Eggplant, or Hot chocolate.

---

### Do you want further details or have any questions?

### 5 Related Questions:
1. How would the result change if Chris added another dessert option, say "Ice cream"?
2. What if Chris wanted to serve exactly **two** of Artichoke, Eggplant, or Hot chocolate?
3. How would the number of possible meals change if they introduced a new main course, "Falafel"?
4. What if Chris could serve both an appetizer and a main course but skipped dessert?
5. How many meals would be possible if Chris had to serve **both** Artichoke and Hot chocolate?

### Tip:
For counting problems like this, consider using systematic lists or diagrams (like tree diagrams) to visualize all options and restrictions.

Chris (they/them) is planning a meal for a party. The meal will include an appetizer, a main course, and dessert. They have three choices for an appetizer (Artichoke, Bruschetta, or Caviar), two choices for a main course (Drumsticks or Eggplant), and three choices for dessert (Fritter, Gelato, or Hot chocolate). b. How many different meals are possible? c. How many different meals are possible if Chris cannot serve Eggplant and Fritter together? d. How many different meals are possible if Chris must serve at least one of Artichoke, Eggplant, or Hot chocolate?

Explore how to calculate the total number of meal combinations for a party menu with specific restrictions using basic combinatorics. This problem involves the Fundamental Counting Principle, subtraction of invalid options, and conditional combinations. Suitable for Grades 7-9.

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