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When we speak about factorial notation in mathematics, we're referring to a shorthand way to express the multiplication of a series of descending natural numbers. Specifically, the factorial of a non-negative integer, let's say, reference number 'n', is the product of all positive integers less than or equal to 'n'. It is denoted as 'n!'.

For example, if you have to write 4 factorial, it would look like this: \( 4! = 4 \times 3 \times 2 \times 1 = 24 \). Factorial notation is incredibly useful in fields such as combinatorics, probability, and algebra to simplify and calculate expressions involving permutations and combinations.

Combinatorics is a branch of mathematics that deals with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many areas of mathematics and has many applications ranging from logic to statistical physics, from evolutionary biology to computer science, among others.

One of the most fundamental problems in combinatorics is determining how many different ways there are to arrange or select certain items. For this, combinatorics utilizes a variety of techniques, concepts, and formulae such as the use of factorial notation, permutations, and combinations to solve related problems.

The term 'n factorial', denoted by the symbol \( n! \), is fundamental in the permutation formula and is a basic concept in combinatorics. The factorial represents the product of all positive integers up to 'n'. It's important to remember that by definition, the factorial of zero, \( 0! \), is 1.

Why is this the case? It's because zero factorial provides a necessary placeholder that allows the formulas of combinatorics and algebra to work consistently for all number ranges. For instance, when you have 'no items' to multiply (as is the case with zero), the multiplication result is 1, thus making \( 0! = 1 \) a universally accepted convention.

The permutation formula is a key combinatorial expression used to find the number of possible arrangements of a subset of items within a larger set. When we want to calculate the permutations of 'n' elements taken 'r' at a time, we use the formula \( P(n,r) = \frac{n!}{(n-r)!} \).

Here's how it works: 'n' is the total number of elements to choose from, and 'r' is the number of elements we are choosing. 'n!' tells us the number of ways to arrange 'n' items, and '(n-r)!' accounts for the fact that the order of the remaining 'n-r' items doesn't matter for our current selection. This formula gives us the ability to calculate permutations quickly and is indispensable when we deal with probability problems and other related scenarios in combinatorics.