# A club of 12 people would like to choose a person for each office of president, a vice president, and a secretary. How many different ways are there to select the officers so that only one person holds each office?

## to solve this problem we shall use Combinatorics

To determine the number of different ways to select a president, a vice president, and a secretary from a group of 12 people where each office is held by a different person, we can consider this a problem of arranging a subset of the group into specific positions.

We start by selecting the president. There are 12 choices for who could be the president. After selecting the president, that person is no longer available for the other offices.

Next, we select the vice president. Since one person has already been chosen as president, there are 11 remaining choices for vice president.

Finally, we select the secretary. With two positions already filled (president and vice president), there are 10 remaining choices for secretary.

The total number of ways to fill these positions is the product of the number of choices for each office:

12×11×10