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The problem shown in the image asks for different ways to select a **president**, **secretary**, and **treasurer** from the set \( N = \{ \text{Adam (A)}, \text{Brad (B)}, \text{Cassie (C)}, \text{Daniel (D)}, \text{Elaine (E)} \} \), where the president must be a junior and the other two (secretary and treasurer) must be seniors.

The key constraints are:
- The president must be chosen from the juniors: \( \{ A, B, D \} \) (Adam, Brad, and Daniel).
- The secretary and treasurer must be chosen from the seniors: \( \{ C, E \} \) (Cassie and Elaine).
- No one can hold more than one office.

We need to list all possible valid combinations where:
- The president is a junior.
- The secretary and treasurer are seniors.
- There are no repeated individuals in any trio.

### Steps to solve:
1. **Choose the president**: From the juniors \( \{A, B, D\} \).
2. **Choose the secretary and treasurer**: From the seniors \( \{C, E\} \), and there are two distinct ways to assign them to the roles of secretary and treasurer: \( \{C, E\} \) or \( \{E, C\} \).

### Possible combinations:
- **President: A** can be paired with:
   - \( C \) as secretary and \( E \) as treasurer (ACE).
   - \( E \) as secretary and \( C \) as treasurer (AEC).
- **President: B** can be paired with:
   - \( C \) as secretary and \( E \) as treasurer (BCE).
   - \( E \) as secretary and \( C \) as treasurer (BEC).
- **President: D** can be paired with:
   - \( C \) as secretary and \( E \) as treasurer (DCE).
   - \( E \) as secretary and \( C \) as treasurer (DEC).

### Valid combinations:
- \( \text{ACE}, \text{AEC}, \text{BCE}, \text{BEC}, \text{DCE}, \text{DEC} \).

Looking at the provided choices:
- **Option C** lists all the valid combinations: ACE, AEC, BCE, BEC, DCE, DEC.

Thus, the correct answer is **C**.

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Would you like any further details or have questions? Here are 5 related questions you might find useful:

1. What is the total number of possible combinations if there were no restrictions on the juniors and seniors?
2. How would the solution change if Cassie and Elaine were not both seniors?
3. How does the solution change if Adam is not eligible for president?
4. What if more offices needed to be filled, such as a vice president role?
5. How would we calculate the probability of randomly selecting a valid trio?

**Tip**: When solving combinatorial problems with restrictions, always list your choices step by step to avoid missing any valid combinations.

Assuming all members are eligible, but no one can hold more than one office, list and count the different ways the club could select a president, a secretary, and a treasurer if the president must be a junior and the other two must be seniors. (Cassie and Elaine are seniors, and the others are juniors.)

Explore how to solve a combinatorial problem involving the selection of a president, secretary, and treasurer from a group of juniors and seniors. Learn step-by-step methods to count valid combinations using combinatorial principles. Suitable for students in Grades 9-11.

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