A certain committee consists of 16 people. From the committee, a president, a vice-president, and a treasurer are to be chosen. In how many ways can these 3 offices be filled? Assume that a committee member can hold at most one of these offices.

To solve this problem, we need to determine how many ways we can choose a president, a vice-president, and a treasurer from a committee of 16 people, where each person can hold only one position.

### Step-by-step solution:

1. **Choosing the president**: 
   - There are 16 people in the committee, so there are 16 choices for who can be the president.

2. **Choosing the vice-president**:
   - After the president is chosen, 15 people remain. So, there are 15 choices for who can be the vice-president.

3. **Choosing the treasurer**:
   - After both the president and the vice-president are chosen, 14 people remain. Therefore, there are 14 choices for who can be the treasurer.

### Total number of ways:
To find the total number of ways to fill the three offices, multiply the number of choices for each position:

\[
16 \times 15 \times 14 = 3360
\]

Thus, the 3 offices can be filled in **3,360 different ways**.

Would you like more details or further explanation?

Here are 5 related questions:
1. How would the number of ways change if there were 20 committee members instead of 16?
2. What if the roles were chosen without regard to the specific office (just selecting 3 people)?
3. How many ways can we choose 4 different roles from the 16 committee members?
4. What is the probability that a specific person is chosen as the president?
5. How does this change if a person can hold more than one office?

**Tip**: When filling distinct positions from a group, the key principle is to multiply the number of remaining choices at each step.

Learn how to determine the number of ways to choose a president, vice-president, and treasurer from a committee of 16 people using combinatorics and permutation formulas. This problem demonstrates key concepts such as the multiplication principle and fundamental counting theorem. Suitable for Grades 8-10.

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