Arrangements (Permutations)
Big Idea: An arrangement is an ordered selection of objects. The key question is: "In how many different ways can we arrange objects when order matters?"
📐 Formal Definition
A permutation (arrangement) is an ordered selection of items from a collection. Two arrangements are different if they contain different items or the same items in a different order.
For example, the arrangements ABC and BAC are considered different because the order is different, even though they use the same letters.
🔑 The Fundamental Counting Principle
If one event can occur in $m$ ways and a second event can occur in $n$ ways, then the two events together can occur in:
$$m \times n \text{ ways}$$
This extends to any number of events: if there are $k$ events with $n_1, n_2, \ldots, n_k$ choices respectively, the total number of outcomes is:
$$n_1 \times n_2 \times n_3 \times \cdots \times n_k$$