Counting Principles

This is a permutation problem. We have 10 choices for the president, 9 choices for the vice president (since one person is already chosen as president), and 8 choices for the secretary. Therefore, the total number of ways is 10 * 9 * 8 = 720.

This is a combination problem. We have 12 choices for the first person, 11 choices for the second person, and so on. Therefore, the total number of ways is 12 * 11 * 10 * 9 * 8 = 792.

This is a combination problem. We have 12 total balls, and we need to choose 3 of them. Therefore, the total number of ways is 12 * 11 * 10 / 3! = 120.

This is a multiplication principle problem. We have 10 * 8 * 6 = 480 ways to fill the engineering positions, 8 * 6 = 48 ways to fill the accounting positions, and 6 ways to fill the marketing manager position. Therefore, the total number of ways to fill all the positions is 480 * 48 * 6 = 2,880.

This is a multiplication principle problem. Each question has 4 possible answers, so the total number of ways to answer all the questions is 4^10 = 1048576.

We can first select 1 red marble in 10 ways, 1 blue marble in 15 ways, and 1 green marble in 20 ways. Then, we can select the remaining 2 marbles in 9 * 8 / 2! = 36 ways. Therefore, the total number of ways is 10 * 15 * 20 * 36 = 1,120.

We can first choose which shelf each book will be on in 10^100 ways. Then, we can arrange the books on each shelf in 10! ways. Therefore, the total number of ways is 10^100 * 10! = 10^90.

We can first select 1 woman in 5 ways. Then, we can select the remaining 2 employees from the remaining 9 employees in 9 * 8 / 2! = 36 ways. Therefore, the total number of ways is 5 * 36 = 150.

We can first select 2 officers in 6 * 5 / 2! = 15 ways. Then, we can select the remaining 3 members from the remaining 9 members in 9 * 8 * 7 / 3! = 168 ways. Therefore, the total number of ways is 15 * 168 = 2,100.

We can first select 3 engineers in 5 * 4 * 3 / 3! = 20 ways. Then, we can fill the remaining 7 positions in 7! ways. Therefore, the total number of ways is 20 * 7! = 20,160.

This is a multiplication principle problem. The customer has 10 choices for the appetizer, 12 choices for the main course, and 8 choices for the dessert. Therefore, the total number of ways is 10 * 12 * 8 = 960.

We can first select 1 woman in 5 ways. Then, we can select the remaining 2 employees from the remaining 9 employees in 9 * 8 / 2! = 36 ways. Therefore, the total number of ways is 5 * 36 = 120.

We can first select 2 engineers in 5 * 4 / 2! = 10 ways. Then, we can select the remaining 8 positions in 8! ways. Therefore, the total number of ways is 10 * 8! = 20,160.

We can first select an appetizer in 10 ways. Then, we can select a main course in 12 ways, but we cannot select the same appetizer again. Therefore, we have 11 choices for the main course. Finally, we can select a dessert in 8 ways, but we cannot select the same main course again. Therefore, we have 7 choices for the dessert. Therefore, the total number of ways is 10 * 11 * 7 = 768.