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Factorial permutations are fundamental in understanding various probability problems, such as team selections or arranging items in a specific order. The concept of factorial, denoted by the exclamation mark (!), represents the product of all positive integers up to a given number. For example, the factorial of 5 (5!) is calculated as 5 × 4 × 3 × 2 × 1, which equals 120.

When it comes to permutations, the factorial of a number dictates the total possible arrangements of a set of distinct objects. Considering our soccer team scenario, with 11 unique positions and 11 different players to choose from, the total possible permutations are represented by 11!. This means that the first position could be any of the 11 players, the second position one of the remaining 10, and so on, until all positions are filled.

This concept is the cornerstone of calculating probabilities in scenarios where the order of selection is important, since each permutation represents a unique outcome. Understanding factorial permutations enables us to determine the size of our sample space in probability calculations, which is the first step in assessing the likelihood of any given event.

The calculation of event probability involves determining the likelihood of a specific event occurring within a set of possible outcomes. To calculate this, we follow a simple formula: the number of favorable outcomes divided by the total number of possible outcomes. The result is a number between 0 and 1, where 0 indicates that the event cannot occur, and 1 represents certainty that the event will occur.

In our exercise dealing with the soccer team selection, the probability calculation for event A needs us to consider only the outcomes where Mabel, Keisha, or Diedra end up as the goalkeeper. Out of the 11! total permutations, there are 3 favorable outcomes for this specific event, since any one of the three could be the goalkeeper. Therefore, the probability is calculated by dividing 3 by 11, leading to a probability of 3/11.

Similarly, for event B and C, additional conditions are added, and the probability is further calculated by multiplying the probabilities of the independent events. For instance, the probability for event B incorporates the probability of event A with the additional probability of either Alice or Phyllis being the center forward, reflecting the interdependent nature of the events.

Combinatorial probability is concerned with the likelihood of an event occurring where the order of selection does not matter, often referred to as combinations. This contrasts with permutations where the order is significant. However, in our soccer team problem, while the specific positions being chosen for each player matters, the events can be treated as combinatorial problems when focusing on smaller groups within the team.

To elaborate, consider the event of choosing a goalkeeper. We are looking at the combination of selecting 1 goalkeeper out of 3 potential candidates, while the specific order in which other team members are chosen does not affect this particular selection. Hence, we can use combinatorial probability, combined with factorial permutations, to calculate event probabilities.

Furthermore, when we have sequences of independent choice events, as seen with the selection of goalkeeper, center forward, and left fullback, we can treat each selection as a separate combinatorial problem and multiply their probabilities to find the combined outcome likelihood. This approach allows us to understand more intricate scenarios where several conditions must be met simultaneously, such as in our extended problem with events B and C.