Combinatorics

This is a combination problem. The formula for combinations is C(n, r) = n! / (n - r)!. In this case, n = 10 and r = 5, so C(10, 5) = 10! / (10 - 5)! = 10! / 5! = 252.

This is a combination problem with restrictions. First, you need to select 1 red ball from 6 red balls, which can be done in 6 ways. Then, you need to select 1 blue ball from 4 blue balls, which can be done in 4 ways. Finally, you need to select 1 green ball from 2 green balls, which can be done in 2 ways. So, the total number of ways is 6 * 4 * 2 = 144.

This is a permutation problem. The formula for permutations is P(n, r) = n! / (n - r)!. In this case, n = 10 and r = 3, so P(10, 3) = 10! / (10 - 3)! = 10! / 7! = 120.

This is a permutation problem with repetitions. The formula for permutations with repetitions is P(n, r) = n^r. In this case, n = 15 and r = 3, so P(15, 3) = 15^3 = 2220.

This is a combination problem with restrictions. First, you need to select 2 red balls from 10 red balls, which can be done in C(10, 2) = 45 ways. Then, you need to select 3 balls from the remaining 27 balls (12 blue balls + 15 green balls), which can be done in C(27, 3) = 2925 ways. So, the total number of ways is 45 * 2925 = 3003.

This is a combination problem with restrictions. First, you need to select the 2 particular people from the group of 10 people, which can be done in C(10, 2) = 45 ways. Then, you need to select 3 more people from the remaining 8 people, which can be done in C(8, 3) = 56 ways. So, the total number of ways is 45 * 56 = 252.

This is a combination problem with restrictions. First, you need to select the 2 particular employees who cannot work together from the group of 12 employees, which can be done in C(12, 2) = 66 ways. Then, you need to select 4 employees from the remaining 10 employees, which can be done in C(10, 4) = 210 ways. So, the total number of ways is 66 * 210 = 495.

This is a combination problem with restrictions. First, you need to select 1 red ball from 8 red balls, which can be done in 8 ways. Then, you need to select 1 blue ball from 6 blue balls, which can be done in 6 ways. Finally, you need to select 1 green ball from 4 green balls, which can be done in 4 ways. So, the total number of ways is 8 * 6 * 4 = 420.

This is a permutation problem without repetitions. The formula for permutations without repetitions is P(n, r) = n!. In this case, n = 10 and r = 3, so P(10, 3) = 10! = 720.

This is a combination problem with restrictions. First, you need to select 2 women from the group of 15 employees, which can be done in C(7, 2) = 21 ways. Then, you need to select 3 more employees from the remaining 13 employees, which can be done in C(13, 3) = 286 ways. So, the total number of ways is 21 * 286 = 1260.

This is a combination problem with restrictions. First, you need to select 1 red ball from 10 red balls, which can be done in 10 ways. Then, you need to select 1 blue ball from 12 blue balls, which can be done in 12 ways. Finally, you need to select 3 more balls from the remaining 27 balls (15 green balls + 10 red balls + 12 blue balls), which can be done in C(27, 3) = 2925 ways. So, the total number of ways is 10 * 12 * 2925 = 4320.

This is a combination problem with restrictions. First, you need to select the 2 particular people who must be on the committee and seated next to each other, which can be done in 2 ways. Then, you need to select 3 more people from the remaining 10 people, which can be done in C(10, 3) = 120 ways. So, the total number of ways is 2 * 120 = 1560.

This is a combination problem with restrictions. First, you need to select the 3 particular employees who cannot work together from the group of 18 employees, which can be done in C(18, 3) = 816 ways. Then, you need to select 6 employees from the remaining 15 employees, which can be done in C(15, 6) = 5005 ways. So, the total number of ways is 816 * 5005 = 6720.

This is a combination problem with restrictions. First, you need to select 2 red balls from 12 red balls, which can be done in C(12, 2) = 66 ways. Then, you need to select 2 blue balls from 10 blue balls, which can be done in C(10, 2) = 45 ways. Finally, you need to select 1 more ball from the remaining 20 balls (8 green balls + 12 red balls + 10 blue balls), which can be done in 20 ways. So, the total number of ways is 66 * 45 * 20 = 2880.