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Understanding permutations and combinations is fundamental when dealing with probability questions, especially when it comes to dice. Permutation involves the arrangement of objects in a specific order, and it is crucial when the order matters. For example, when we want to know how many ways we can rearrange a set of unique dice faces.

In contrast, a combination is a selection of items from a larger pool where the order does not matter. If we were considering combinations in our dice problem, we would be looking at how many ways we could choose a subset of dice faces, disregarding the sequence in which they appear. However, since our dice problem requires each die to show a different face, we use permutations to solve it.

The concept of factorial is instrumental in calculating permutations. Given a number n, its factorial is the product of all positive integers less than or equal to n and is denoted by n!. Factorials grow extremely fast with larger numbers, indicating the swift increase in the number of permutations as more elements are involved.

In dice probability problems, calculating the factorial allows us to determine the total number of possible arrangements for a set of dice. For instance, the factorial of 6, noted as 6!, calculates all the ways to arrange six dice, resulting in 720 unique permutations.

Probability calculation is the process of determining the chance of a particular event occurring. It's generally expressed as a fraction or a percentage, with the probability of an event A noted as P(A). The basic formula is P(A) = (number of favorable outcomes) / (total number of possible outcomes).

To accurately compute probability, one must first understand how to count both the favorable and total outcomes correctly. This is where factorial, permutation, and combination principles become highly valuable. They provide systematic methods to count outcomes, thereby ensuring an accurate calculation of probability.

Dice probability problems often involve calculating the likelihood of various outcomes when one or more dice are rolled. The complexity can vary from simple scenarios, like the probability of rolling a single six with one die, to more complex ones, such as our textbook example where all six faces must appear once when six dice are rolled.

A critical step in solving these problems is distinguishing between the total number of outcomes (which, for one die, is always 6) and the number of favorable outcomes (which varies depending on the specific event). With this distinction clear, we're equipped to use the methods of factorial and permutation to calculate probabilities accurately in these nuanced scenarios.