You are forming 3 teams from 25 people.
Team A has 8
Team B has 3
Team C has 14
How many ways can team A be selected?
How many ways can team B selected from the remaining people?
How many ways can team C be selected from the remaining people?
How many ways can all teams be selected?
Whats the probability that 3 random people selected would all be from team A?
To calculate the number of ways each team can be selected, we can use the concept of combinations.
1. Number of ways team A can be selected:
To form Team A, we need to choose 8 people out of the 25. Therefore, the number of ways Team A can be selected is given by the combination formula:
C(25, 8) = 25! / (8!(25-8)!) = 1081575
2. Number of ways team B can be selected from the remaining people:
After forming Team A, we are left with 25 - 8 = 17 people. To form Team B, we need to choose 3 people out of these 17. Thus, the number of ways Team B can be selected is:
C(17, 3) = 17! / (3!(17-3)!) = 680
3. Number of ways team C can be selected from the remaining people:
Once Teams A and B are formed, we now have 17 - 3 = 14 people remaining. To form Team C, we need to choose 14 people out of these 14 remaining. So, the number of ways Team C can be selected is:
C(14, 14) = 14! / (14!(14-14)!) = 1
4. Number of ways all teams can be selected:
The total number of ways all teams can be selected is the product of the number of ways each team can be selected:
Total number of ways = Number of ways Team A can be selected * Number of ways Team B can be selected * Number of ways Team C can be selected
= 1081575 * 680 * 1
= 736,926,000
5. Probability of selecting 3 random people from Team A:
Since there are 8 people in Team A and 25 people in total, the probability of randomly selecting 3 people from Team A is:
P(3 people from Team A) = C(8, 3) / C(25, 3)
= (8! / (3!(8-3)!)) / (25! / (3!(25-3)!))
= (56 / 10,626) ≈ 0.00527
Therefore, the probability that 3 randomly selected people are all from Team A is approximately 0.00527.
To answer these questions, we can use the concept of combinations.
1. How many ways can team A be selected?
Since there are 8 people in team A, we need to choose 8 people from a total of 25. We can calculate this using the formula for combinations: nCr = n! / (r!(n - r)!), where n is the total number of people and r is the number of people to be selected. In this case, it would be 25C8 = 25! / (8!(25-8)!) = 25! / (8!17!).
2. How many ways can team B be selected from the remaining people?
After selecting 8 people for team A, we now have 25 - 8 = 17 people remaining. We need to choose 3 people from this group for team B. So we can calculate this as 17C3.
3. How many ways can team C be selected from the remaining people?
After selecting teams A and B, we now have 25 - 8 - 3 = 14 people remaining. We need to choose all 14 of them for team C. So we can say that there is only one way to select team C from the remaining people.
4. How many ways can all teams be selected?
To find the total number of ways to select all teams, we multiply the number of ways to select each team. So, the total number of ways is given by: 25C8 * 17C3 * 1.
5. What's the probability that 3 random people selected would all be from team A?
To find the probability, we need to calculate the number of favorable outcomes (3 people selected from team A) and divide it by the total number of possible outcomes (any 3 people selected from the total number of people). The probability would be: (number of ways to select 3 people from team A) / (total number of ways to select any 3 people from 25). It can be calculated as: 8C3 / 25C3.