Counting techniques

Counting techniques are fundamental methods we use in combinatorics to figure out how many possibilities exist in a given situation.

 

We need counting techniques because it quickly becomes unmanageable to count all possibilities manually when the numbers grow. With the help of general rules, we can calculate the number of combinations systematically and reliably.

 

A simple example is a password with three digits. Instead of writing out all possibilities (000, 001, 002 etc.), we can use a counting technique: each digit has 10 possibilities, and there are three digits in total. Thus:

 

$$ \large 10 \cdot 10 \cdot 10 = 1000\ possibilities $$

 

Four fundamental techniques

The most commonly used counting techniques are:

 

• Addition method: When you can choose either one or the other.
• Multiplication method: When you need to choose several things at the same time.
• Division rule: When there are repetitions, and we must avoid counting the same thing multiple times.
• Inclusion–exclusion principle: When sets overlap, and we need to correct for double counting.

 

These techniques are the building blocks of combinatorics and form the basis for topics such as permutations, combinations and much more.