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Organize cases systematically

Organizing cases systematically is the ability to represent all possible outcomes of a finite situation in an orderly, nonoverlapping, and complete way, using sorted lists, tables, or tree diagrams to avoid omissions and double-counting. It includes distinguishing whether order matters and counting simple combinations of choices, while not extending to formal combinatorial formulas or highly generalized counting methods; this reasoning supports probability, proportional reasoning, and algebraic analysis of patterns.

Detailed Explanation: 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 AA, BB, CC, and DD. How many codes are possible?

Because a code has a first letter and a second letter, order matters. For example, ABAB and BABA are different codes.

Group the codes by their first letter:

First letterPossible second lettersCodes
AAB,C,DB,C,DAB,AC,ADAB, AC, AD
BBA,C,DA,C,DBA,BC,BDBA, BC, BD
CCA,B,DA,B,DCA,CB,CDCA, CB, CD
DDA,B,CA,B,CDA,DB,DCDA, DB, DC

Each row has 33 codes, and there are 44 rows:

3+3+3+3=123+3+3+3=12

So, there are 1212 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.

Learn by doing: Organize cases systematically

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