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| Combinatorics | |
| Product rule | |
| Sum Rule | |
| Permutations | |
| Combinations | |
| Arrangement | |
| Repetitions | |
| Mutually exclusive | |
| The order of the articles |
Introduction
Combinatorics is the study of collections of objects. Specifically, counting objects, arrangement, derangement, etc. of objects along with their mathematical properties. Counting objects is important in order to analyze algorithms and compute discrete probabilities. Originally, combinatorics was motivated by gambling: counting configurations is essential to elementary probability.A simple example: How many arrangements are there of a deck of 52 cards? In addition, combinatorics can be used as a proof technique. A combinatorial proof is a proof method that uses counting arguments to prove a statement.
Product rule
If two events are not mutually exclusive (that is, we do them separately), then we apply the product rule.
Theorem (Product Rule)
Suppose a procedure can be accomplished with two disjoint subtasks. If there are 
ways of doing the first task and 
ways of doing the second, then there are
ways of doing the overall procedure.
Example 1
You are trying to construct a schedule. You have 3 courses you could take at 8:30, and 2 courses you could take at 9:30. How many different possible schedules are there?
Solution
The answer is the product of 3 and 2, or 6. It is easy to see how this rule is derived, if you simply produce a diagram of the possibilities. Suppose that the available courses at 8:30 are labeled A, B, and C, and the available courses at 9:30 are D and E. You obtain 6 possible pathways, as
shown on the following picture.
Sum Rule
If two events are mutually exclusive, that is, they cannot be done at the same time, then we must apply the sum rule.
Theorem (Sum Rule)
If an event 
can be done in ways and an event 
can be done in 
ways and and 
are mutually exclusive, then the number of ways of both events occurring is
Example
If there are 3 different courses offered in the morning and 2 different course offered in the afternoon. There will be 3 + 2 choices for a student to enroll in only one course.
