A club with sixteen members is to choose three​ officers: ​ president, vice-president, and​ secretary-treasurer. If each office is to be held by one person and no person can hold more than one​ office, in how many ways can those offices be​ filled?
Question content area bottom
Part 1
  
enter your response here ways

To solve this problem, we need to calculate the number of ways to choose 3 officers (president, vice-president, and secretary-treasurer) from a group of 16 members, where no one can hold more than one office.

Since the order in which we assign the offices matters (different people in different roles count as different arrangements), this is a **permutation** problem.

### Steps:
1. **Choose the president**: There are 16 people to choose from.
2. **Choose the vice-president**: After selecting the president, there are 15 people remaining to choose the vice-president.
3. **Choose the secretary-treasurer**: After selecting the president and vice-president, 14 people remain for this position.

Now, multiply the number of choices for each office:
\[
16 \times 15 \times 14 = 3360
\]

Thus, there are **3,360 ways** to fill these three offices.

Would you like more details or have any questions?

Here are 5 related questions for practice:
1. How many ways can a committee of 3 be selected from the 16 members, if the order doesn't matter?
2. If the club added another officer role, how would this affect the total number of arrangements?
3. What if the club decided that the president and vice-president could switch roles after selection? How would this affect the calculation?
4. If there were only 10 members in the club, how many ways could the three offices be filled?
5. What if one person is ineligible to be president? How would that change the number of possible outcomes?

**Tip:** Always identify whether a problem involves combinations (order doesn't matter) or permutations (order matters).

A club with sixteen members is to choose three officers: president, vice-president, and secretary-treasurer. If each office is to be held by one person and no person can hold more than one office, in how many ways can those offices be filled?

Calculate the number of ways to choose three officers (president, vice-president, and secretary-treasurer) from 16 members, using permutation concepts. This problem involves key combinatorics and permutation formulas, suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation