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Factorial notation is fundamental to understanding permutations in statistics. It's signified by an exclamation point (!) following a number, and represents the product of all positive integers up to that number. For instance, the factorial of 6, written as 6!, is calculated as
\(6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720\).
Factorials grow very fast with larger numbers, which indicates just how quickly the number of permutations can increase. It's critical for calculations where order matters, such as arranging objects or choosing sequential events. When considering permutations, the factorial denotes the total number of ways a set number of items can be ordered.

Probability is a way of quantifying the likelihood that a given event will occur. It's expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. The probability P of an event is calculated by dividing the number of favorable outcomes by the total number of possible outcomes:
\( P(\text{event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \).
Understanding the basic principle of probability allows us to tackle more complex problems, such as calculating permutation probabilities. It's essential to consider both the favorable and total outcomes, bearing in mind that all outcomes must be equally likely for the probability calculation to be valid.

Permutation probability combines the concepts of factorial notation and probability to determine how likely a specific arrangement of items is among all possible permutations. In a situation where order is important, like in the example of the six-letter word NIMBLE, it helps to calculate the probability that a permutation will have a particular form or characteristic.
To improve understanding, consider the permutations that begin with the letter M. Since M is fixed at the beginning, the remaining letters can be arranged in 5! ways as detailed in our exercised solution. The probability, then, of getting a permutation that starts with M is the ratio of favorable permutations (those beginning with M) to total permutations (all possible arrangements). Using our factorial method, this becomes \(\frac{5!}{6!}\), by simplifying this, we see the six cancels out, leaving us with \(\frac{1}{6}\) or about 16.67% when converted to a percent.
Overall, permutation probability is invaluable when needing to calculate the likelihood of different arrangements or sequences in a set of items.