Answer:
To determine the probability that a randomly chosen team of 3 players from the hat does not contain certain players (e.g., the captain and the vice-captain), you can use the concept of combinations and probabilities.
First, let's calculate the total number of ways to choose a team of 3 players from the hat, which is a combination problem. You have 7 players in the hat, and you want to choose 3 of them:
Total ways to choose a team of 3 from 7 = C(7, 3)
C(7, 3) represents the number of combinations, and it can be calculated as:
C(7, 3) = 7! / (3!(7 - 3)!)
C(7, 3) = 7! / (3! * 4!)
C(7, 3) = (7 * 6 * 5) / (3 * 2 * 1) = 35
So, there are 35 different ways to choose a team of 3 players from the hat.
Now, let's consider the probability that the chosen team does not contain the captain and the vice-captain.
There are 2 players (captain and vice-captain) that you want to exclude from the team. To calculate the number of ways to choose a team of 3 players without including the captain and vice-captain, you need to choose 3 players from the remaining 5 players (since 7 - 2 = 5).
Ways to choose a team of 3 players from the remaining 5 = C(5, 3)
C(5, 3) can be calculated as:
C(5, 3) = 5! / (3!(5 - 3)!)
C(5, 3) = 5! / (3! * 2!)
C(5, 3) = (5 * 4) / (2 * 1) = 10
So, there are 10 different ways to choose a team of 3 players from the remaining 5 players without including the captain and vice-captain.
Now, you can calculate the probability that the chosen team does not contain the captain and vice-captain:
Probability = (Number of ways to choose a team without captain and vice-captain) / (Total number of ways to choose a team)
Probability = 10 / 35
Simplify the fraction if necessary:
Probability = 2/7
So, the probability that a randomly chosen team of 3 players does not contain the captain and vice-captain is 2/7.
Explanation: