Permutations and Combinations - Fundamental Principle of Counting

The Fundamental Principle of Multiplication states that if an event can occur in $m$ different ways, following which another event can occur in $n$ different ways, then the total number of ways of occurrence of both events in the specified order is $m \times n$. This can be visualized as a rectangular grid where the $m$ rows represent choices for the first event and $n$ columns represent choices for the second event, with each intersection representing a unique outcome.

The Fundamental Principle of Addition applies when two tasks are mutually exclusive. If one task can be performed in $m$ ways and another task can be performed in $n$ ways, then either of the two tasks can be performed in $m + n$ ways. Visually, this is like looking at two separate, non-overlapping circles (sets) and counting the total number of items across both.

Factorial Notation ($n!$) represents the product of the first $n$ consecutive natural numbers. For any positive integer $n$, $n! = n \times (n-1) \times (n-2) \times \dots \times 3 \times 2 \times 1$. By convention, $0! = 1$. This can be visualized as a descending ladder of numbers being multiplied together.

The 'Box Method' is a visual strategy used to solve counting problems. We draw a series of empty boxes, where each box represents a position or a sub-task. We then fill each box with the number of available choices for that position and multiply the numbers together to find the total combinations.

Tree Diagrams are visual tools used to list all possible outcomes of a sequence of events. Starting from a single point, branches are drawn for each choice of the first event; from the end of those branches, new branches are drawn for the choices of the second event, creating a branching structure that resembles a tree.

Repetition vs. Non-repetition: In many counting problems, it is crucial to identify if items can be reused. If repetition is allowed, the number of choices remains constant for each step; if repetition is not allowed, the number of choices decreases by 1 for each subsequent step, visualized as a shrinking pool of available objects.

Order of operations: When applying the Fundamental Principle of Counting, the order in which choices are made matters if the events are dependent. We always fill the positions with the most restrictive conditions first (e.g., if a number must be even, fill the units place first).