Given N people, the number of different ways to choose, for example, an n-person committee where the order does matter (aka, each person in the committee has a distinct position) would be the total number of repetitions allowed sets divided by the factorial of the number of objects minus the number of trials (see formula).

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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

Ordering and Replacement

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

Probability for the Enthusiastic Beginner by David Morin

If there are N possible results each time a trial occurs, then the total number of possible outcomes where the order of the objects matters is N^n. Examples of this situation include rolling an N-sided die n times, or drawing one of N balls from a box n times and returning the ball to the box each time (so that all of the draws are equivalent).

Ordered sets, repetitions allowed

Ordered sets, repetitions not allowed

Given N people, the number of different ways to choose, for example, an n-person committee where the order doesn’t matter (i.e., everyone has the same interchangeable role) is denoted by the binomial coefficient and is the same formula as ordered sets repetitions not allowed, but includes division by the factorial of n to account for repeated permutations (see formula).

Unordered sets, repetitions not allowed

This is helpful when you want the binomial coefficient of a set of objects where the objects or unordered and repetitions are allowed.

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