Combinatorics

Combinatorics is the study of counting, arranging, and selecting objects, often using systematic methods. It includes topics such as permutations (ordered arrangements), combinations (unordered selections), and the use of the addition and multiplication principles.

Important properties

• Addition Principle: If there are $m$ ways to do one thing and $n$ ways to do another, and they cannot both happen, there are $m + n$ ways to choose one.

• Multiplication Principle: If there are $m$ ways to do one thing and $n$ ways to do another, and both happen in sequence, there are $m \times n$ ways to do both.

• Permutations: The number of ways to arrange $n$ distinct objects is $n!$.

• Combinations: The number of ways to choose $k$ objects from $n$ is $\binom{n}{k}$.