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In probability, a "simple event" refers to a single outcome of an experiment. When you perform an experiment, like tasting different colas, each possible result is a simple event. In the given exercise, the simple events are the different ways the cola glasses can be ranked by preference. Every arrangement has an equal chance of occurring because all glasses contain the identical cola.

This leads us to deduce the simple events as follows:

• C, D, P - Cola C is preferred over D and P.
• C, P, D - Cola C is preferred over P and D.
• D, C, P - Cola D is preferred over C and P.
• D, P, C - Cola D is preferred over P and C.
• P, C, D - Cola P is preferred over C and D.
• P, D, C - Cola P is preferred over D and C.

Understanding simple events is crucial, especially when it comes to calculating the probability of each outcome. Here, each simple event has a probability of being chosen as \( \frac{1}{6} \) because there are six equally likely orderings.

Permutations are an arrangement of items in a specific order. When the order matters, as is the case with ranking items, permutations play a significant role. The more items there are to arrange, the more potential permutations exist. For the glasses of cola labeled "C," "D," and "P," understanding permutations enables us to determine all possible rankings.

Permutations depend on the number of items being arranged. Here, there are three glasses, leading to many possible orderings of the glasses—each is considered a permutation. The mathematical notation for a permutation of three distinct items is 3! (3 factorial), which represents all possible orderings:

• First, position option: choose any glass (3 options).
• Second position option: choose from the remaining glasses (2 options).
• Final position option: only one glass left (1 option).

When you multiply these choices, you get \(3 \times 2 \times 1 = 6\) permutations. Thus, in our exercise, there are six possible rankings.

The factorial, often noted with an exclamation mark (!), is a mathematical operation that multiplies a number by all the positive integers below it. It's a way of determining the total arrangements of a set of objects.

For example, 3! is shorthand for:
\[ 3! = 3 \times 2 \times 1 = 6 \]This formula is crucial for finding permutations where order matters, such as ordering our cola glasses in different ways. Whenever you want to calculate how many ways to arrange a set, factorial is your go-to function.

Factorials grow quickly, which means for even a slightly larger number of objects, the number of permutations surges significantly. In our cola example, understanding how factorials work explains why, despite having only 3 glasses, there are exactly 6 different possible simple events to consider. Familiarity with the factorial function can help simplify many probability problems where arranging items is key.