Question: In how many different ways can a club with 10 members select a President, Secretary, and Treasurer, if no member can hold two offices and each member is eligible for any office? Explain in detail.

Answer:

To solve this problem, we need to apply the concept of permutations. Permutations are arrangements of objects in a specific order. In this case, we want to find the number of ways to select a President, Secretary, and Treasurer from 10 members of the club.

Step 1: Finding the number of choices for each position

Since no member can hold two offices and each member is eligible for any office, the number of choices for each position can be calculated as follows:

- President: We have 10 members to choose from, so there are 10 choices for the President position.
- Secretary: Once the President is chosen, there are 9 remaining members to choose from for the Secretary position.
- Treasurer: Once the President and Secretary are chosen, there are 8 remaining members to choose from for the Treasurer position.

Step 2: Calculating the total number of ways

To calculate the total number of ways, we need to multiply the number of choices for each position. This can be done using the multiplication principle.

Total number of ways = Number of choices for President * Number of choices for Secretary * Number of choices for Treasurer

Total number of ways = 10 * 9 * 8

Total number of ways = 720

Therefore, there are 720 different ways for the club to select a President, Secretary, and Treasurer from 10 members.

Conclusion:

In conclusion, a club with 10 members can select a President, Secretary, and Treasurer in 720 different ways, given that no member can hold two offices and each member is eligible for any office. This calculation is based on the concept of permutations and the multiplication principle.