A factorial (denoted with an exclamation point N!) is a quantity calculated by taking the product of the first N integers of an integer N. 0! is defined to be 1. 

Factorials

Given a set of N objects, N! (N factorial) is defined to be the product of the first N integers. The number of different permutations of N objects (that is, the number of different ways of ordering them) is N!. In this scenario, the number of objects in the set N is equivalent to the number of trials n. 

University of Texas at Austin

A permutation is a way of ordering a set of objects or rearranging an already ordered set of objects. Important topics concerning permutations include: 
- Ordering and Replacement
- Permutations of N objects

Permutations 

Morin, D. (2016). Probability for the Enthusiastic Beginner. Las Vegas, Nevada: CreateSpace Independent Publishing Platform.

Probability for the Enthusiastic Beginner by David Morin

Permutations can either be repetitions allowed or repetitions not allowed, as well as ordered or unordered. "With replacement" or "repetitions allowed" means the same thing as "identical trials." Examples include rolling dice, flipping coins, and drawing balls from a box with replacement. "Without replacement" or "repetitions not allowed" means the same thing as "depleting trials," or trials where the number of possible outcomes decreases by one after each trial. Examples include picking committees of people (because a given person cannot be repeated), assigning people to seats,  and drawing balls from a box and not replacing them.

As far as "ordered versus disordered sets, ordered objects can be seen as viewed in a line, and disordered objects can be envisioned as an amorphous group. For example, an ordered committee is one where all of the members have given roles such as president or vice president, whereas an unordered committee is one where all roels are equal.

Calculations of the amount of potential permutations for a given set of objects will vary based on the ordering and replacement of the objects in a set. We will cover them all:
- Ordered sets, repetitions allowed
- Ordered sets, repetitions not allowed
- Unordered sets, repetitions not allowed 
- Unordered sets, repetitions allowed

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