# Counting

In order to establish the probability of an event occurring, we must first count the number of possible outcomes (the domain).  The number of possible outcomes depends on whether the set must be ordered, and if duplicates are allowed.

## Permutations

Permutations are different ways of ordering a set of objects.

For instance, how many different ways are there to sit 5 people.   The first person to be seated has 5 choices, the next one 4, then 3, 2, and finally the last person only has one chair left to choose from.  Therefore the number of permutations are $5*4*3*2*1$.

Or, in generalized form, the number of possible permutations of a set of $N$ objects is given by:

$P_N=N!$.

## Ordered Sets, Repetitions Allowed

For instance, how many outcomes or a toss of a coin, or a roll of a die are possible:

$outcomes=N^n$

where $N$ is the number of possible outcome of each event (2 for the flip of a coin, 6 for a roll of a die), and $n$ is the number of events (number of times a coin is tossed or a die rolled).

## Ordered Sets, Repetitions Not Allowed

For instance how many ways of creating a 2 persons committee from a group of 5 people and the ordering is important (i.e. AB is different from BA) but the same person cannot have 2 seats on the committee (therefore AA is not allowed).

$outcomes=\frac{N!}{(N-n)!}$

Where $N$ is the size of the group we choose from (5), and $n$ is the number of elements selected from the set (2).

## Unordered Sets, Repetitions Not Allowed

For instance how many ways of listing a 2 persons committee from a group of 5 people and the ordering is not important (i.e. AB and BA are equivalent).

$outcomes=_NC_n=\Big( ^N_n \Big)=\frac{N!}{n!(N-n)!}$

Where $N$ is the size of the group we choose from (5), and $n$ is the number of elements selected from the set (2).