Skill: Organize cases systematically

To organize cases systematically:

1. Decide what makes two outcomes different.
2. Choose an order for listing the cases.
3. Group cases by one feature, such as the first choice.
4. List every possibility in each group, without repeats.
5. Check that no possibilities were missed.

Example: A two-letter code uses two different letters from $A$, $B$, $C$, and $D$. How many codes are possible?

Because a code has a first letter and a second letter, order matters. For example, $AB$ and $BA$ are different codes.

Group the codes by their first letter:

First letter	Possible second letters	Codes
$A$	$B,C,D$	$AB, AC, AD$
$B$	$A,C,D$	$BA, BC, BD$
$C$	$A,B,D$	$CA, CB, CD$
$D$	$A,B,C$	$DA, DB, DC$

Each row has $3$ codes, and there are $4$ rows:

$$3+3+3+3=12$$

So, there are $12$ possible codes.

The list is systematic because every code starts in exactly one row, and each possible second letter is listed once in that row. This prevents both missing a code and counting the same code twice.