Combinations and Permutations

The number of ways to choose the president is 12. Once the president is chosen, the number of ways to choose the vice-president is 11. Finally, the number of ways to choose the secretary is 10. Therefore, the total number of ways to choose the president, vice-president, and secretary is 12 * 11 * 10 = 1320.

First, select 1 red ball from 6 red balls, which can be done in 6 ways. Then, select 1 blue ball from 4 blue balls, which can be done in 4 ways. Finally, select 1 green ball from 2 green balls, which can be done in 2 ways. Therefore, the total number of ways to select 3 balls, one of each color, is 6 * 4 * 2 = 48. Since there are 3 different colors, there are 3 different ways to choose which color ball to select first. Therefore, the total number of ways to select 3 balls, at least one of each color, is 3 * 48 = 144.

Since 2 particular people must be included in the committee, we need to choose 3 more people from the remaining 8 people. This can be done in 8C3 = 56 ways. Therefore, the total number of ways to form the committee is 56 * 2 = 112.

Since there must be an equal number of red, blue, and green marbles, we need to select 2 marbles from each color. The number of ways to select 2 red marbles from 10 red marbles is 10C2 = 45. The number of ways to select 2 blue marbles from 15 blue marbles is 15C2 = 105. The number of ways to select 2 green marbles from 20 green marbles is 20C2 = 190. Therefore, the total number of ways to select 6 marbles, 2 of each color, is 45 * 105 * 190 = 891000.

Since 2 particular employees must be included in the team, we need to choose 1 more employee from the remaining 8 employees. This can be done in 8C1 = 8 ways. Therefore, the total number of ways to select the team is 8 * 2 = 16.

First, select 1 red ball from 6 red balls, which can be done in 6 ways. Then, select 1 blue ball from 4 blue balls, which can be done in 4 ways. Now, we have 2 balls left to select, and we can choose either 2 green balls or 1 green ball and 1 red ball or 1 green ball and 1 blue ball. The number of ways to select 2 green balls from 2 green balls is 2C2 = 1. The number of ways to select 1 green ball and 1 red ball from 2 green balls and 5 red balls is 2C1 * 5C1 = 10. The number of ways to select 1 green ball and 1 blue ball from 2 green balls and 3 blue balls is 2C1 * 3C1 = 6. Therefore, the total number of ways to select 4 balls, including at least 1 red ball and at least 1 blue ball, is 6 * 4 * (1 + 10 + 6) = 240.

First, select the president from 15 members, which can be done in 15 ways. Since the president cannot also be the vice-president, there are 14 members left to choose from. Select the vice-president from these 14 members, which can be done in 14 ways. Now, select the secretary from the remaining 13 members, which can be done in 13 ways. Finally, select the treasurer from the remaining 12 members, which can be done in 12 ways. Therefore, the total number of ways to choose the president, vice-president, secretary, and treasurer is 15 * 14 * 13 * 12 = 30030.

Since 3 particular people must be included in the committee, we need to choose 2 more people from the remaining 9 people. This can be done in 9C2 = 36 ways. Therefore, the total number of ways to form the committee is 36 * 2 = 72.

Let x be the number of blue marbles to be selected. Then, the number of red marbles to be selected is 2x, and the number of green marbles to be selected is 3x. Since we are selecting a total of 6 marbles, we have x + 2x + 3x = 6, which gives x = 1. Therefore, we need to select 1 blue marble, 2 red marbles, and 3 green marbles. The number of ways to select 1 blue marble from 15 blue marbles is 15C1 = 15. The number of ways to select 2 red marbles from 10 red marbles is 10C2 = 45. The number of ways to select 3 green marbles from 20 green marbles is 20C3 = 1140. Therefore, the total number of ways to select 6 marbles, with twice as many red marbles as blue marbles and three times as many green marbles as blue marbles, is 15 * 45 * 1140 = 777000.

To divide the group into two teams of 5 people each, we can first choose 5 people for the first team from the 10 people. This can be done in 10C5 = 252 ways. Once the first team is chosen, the second team is automatically determined. Therefore, the total number of ways to divide the group into two teams of 5 people each is 252.

First, select the president from 12 members, which can be done in 12 ways. Once the president is chosen, there are 11 members left to choose from. Select the vice-president from these 11 members, which can be done in 11 ways. Now, select the secretary from the remaining 10 members, which can be done in 10 ways. Finally, select the treasurer from the remaining 9 members, which can be done in 9 ways. Therefore, the total number of ways to choose the president, vice-president, secretary, and treasurer is 12 * 11 * 10 * 9 = 11880.

Since there must be an equal number of red, blue, and green balls, we need to select 2 balls from each color. The number of ways to select 2 red balls from 8 red balls is 8C2 = 28. The number of ways to select 2 blue balls from 6 blue balls is 6C2 = 15. The number of ways to select 2 green balls from 4 green balls is 4C2 = 6. Therefore, the total number of ways to select 6 balls, 2 of each color, is 28 * 15 * 6 = 2520.

Since 2 particular people must not be included in the committee, we need to choose 5 people from the remaining 8 people. This can be done in 8C5 = 56 ways. Therefore, the total number of ways to form the committee is 56.