https://file.aiquickdraw.com/mathbot/file/17297814652056mnwmqcv.jpg

Let's solve this problem step-by-step.

### Problem Breakdown:
- **9 appetizers**, **3 main courses**, and **12 desserts** are available.
- The customer selects:
  - **5 appetizers**
  - **2 main courses**
  - **6 desserts**

The task is to determine the total number of ways the customer can make these selections.

### Step-by-Step Calculation:
Since the order in which the items are chosen doesn't matter, and no item can be selected more than once, we use the **combination formula**:

\[
C(n, r) = \frac{n!}{r! \cdot (n - r)!}
\]

where \(n\) is the total number of items, and \(r\) is the number of items to be selected.

1. **Choosing 5 appetizers from 9:**
   \[
   C(9, 5) = \frac{9!}{5! \cdot (9 - 5)!} = \frac{9 \cdot 8 \cdot 7 \cdot 6}{4 \cdot 3 \cdot 2 \cdot 1} = 126
   \]

2. **Choosing 2 main courses from 3:**
   \[
   C(3, 2) = \frac{3!}{2! \cdot (3 - 2)!} = \frac{3 \cdot 2}{2 \cdot 1} = 3
   \]

3. **Choosing 6 desserts from 12:**
   \[
   C(12, 6) = \frac{12!}{6! \cdot (12 - 6)!} = \frac{12 \cdot 11 \cdot 10 \cdot 9 \cdot 8 \cdot 7}{6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1} = 924
   \]

### Total Number of Ways:
Since the selections are independent, we multiply the number of ways to choose each category.

\[
\text{Total Ways} = C(9, 5) \times C(3, 2) \times C(12, 6)
\]

\[
\text{Total Ways} = 126 \times 3 \times 924 = 349272
\]

### Final Answer:
The total number of ways to select the items is:

\[
\boxed{349272}
\]

Let me know if you have further questions or need a detailed breakdown of any part!

---

Here are 5 related questions for further exploration:
1. How does the combination formula differ from permutations?
2. What if the items could be chosen with replacement? How would the calculation change?
3. How many ways can the customer choose exactly 1 item from each category?
4. What is the probability of randomly selecting a particular combination out of all possibilities?
5. How does factorial growth impact the number of possible selections?

**Tip:** When working with combinations, always remember that the order of selection does not matter. Use permutations only if order is significant.

A catering service offers 9 appetizers, 3 main courses, and 12 desserts. A customer is to select 5 appetizers, 2 main courses, and 6 desserts for a banquet. In how many ways can this be done?

Learn how to solve a combinatorics problem involving selecting 5 appetizers, 2 main courses, and 6 desserts from a catering service menu. This problem uses the combination formula and the fundamental principle of counting, making it suitable for Grades 9-12 students.

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