Counting techniques loosely go under the name of combinatorics. Counting the Total Number of things is called enumerative combinatorics, while counting the maximum, minimum, or minimally sufficient number of things goes by the name of Extremal Combinatorics. Trivial examples are counting the number of ways to roll "18" on four 6-sided die, counting the number of ways to obtain a full house when playing poker with a standard deck, or counting the number of ways to distribute 17 doughnuts of different types to 5 people.
Algebra (see General Discussion)
1. Let's say I roll 6 dice. What is the probability I roll the number ?
2. Counting the number of ways to order 8 couples around a (round) table such that no couple is sitting together.
3. Counting the number of ways to write the integer as the sum of positive integers (e.g. or ).
4. Counting the number of strings of length 8 that can be made from the letters , , and , such that no string has more than two repeating letters (in a row.)
5. Finding the largest integer not of the form where are non-negative integers.
6. Counting the number of ways to color a graph with colors.
7. Counting the maximum number of edges in a graph that does not have a clique of order .
Combinatorics is a field of applied mathematics that can be approached in numerous ways. The standard introduction requires a basic knowledge of algebra, including the factorial function. For a more advanced introduction that may use recurrence relations, basic linear algebra could be useful. Advanced combinatorial tricks tend to use generating functions to transform a recurrence equation into an algebraic or differential equation, and then solve those.