Systematic listing and the product rule explained
Number • Structure and calculation
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Systematic listing - definition
Systematic listing arranges all possible outcomes in an orderly structure so that none are overlooked or repeated. The arrangement uses consistent rules such as rows and columns, tables, ordered pairs, or tree diagrams. The list must include every distinct outcome exactly once to be valid. A correct systematic list uses a clear pattern that allows checking for completeness. The structure makes it straightforward to compare counts from the list with counts from calculation methods.
Common listing strategies
Tables organise choices into rows and columns when two attributes vary independently, so each cell represents one unique outcome. Tree diagrams branch for each choice in sequence, making dependencies visible and showing the total number of outcomes by counting end nodes. Ordered pairs or triples record outcomes as sequences, which help to confirm that all possibilities are present. Systematic ordering prevents duplication by fixing an order for components (for example alphabetical or numerical). The choice of strategy depends on problem size: small problems suit full listing; larger problems use the product rule or mixed methods to avoid lengthy lists.
Product rule - formal statement
The product rule states that when a process consists of two sequential tasks, where the first task can be done in m ways and for each way the second task can be done in n ways, the total number of possible outcomes equals m × n. The rule extends to more than two tasks: if tasks have counts m1, m2, ..., mk and each choice for earlier tasks does not change the number of choices for later tasks, the total is m1 × m2 × ... × mk. The product rule requires that the number of options for each task is fixed given the preceding choices, or that any dependence is accounted for explicitly in the multiplication factors.
When the product rule applies and limitations
The product rule applies when tasks are sequential and the number of choices for each task is independent of the particular choices made earlier, or when any dependence is captured by separate factors. Dependence that changes the number of options invalidates a direct m × n calculation unless the dependence is split into separate cases and counted individually. Order matters in the product rule when tasks represent ordered stages. The product rule counts ordered outcomes. For problems where order does not matter, the product rule may overcount and need adjustment (for example by dividing by permutations or using combinations).
Connection to permutations and combinations
Permutations count ordered arrangements and often use the product rule with changing numbers of options (for example arranging k items from n gives n × (n−1) × ...). Combinations count unordered selections and require division by factorials to remove order; the product rule alone counts ordered sequences and must be adjusted for unordered problems. Clear distinction between ordered and unordered outcomes prevents misapplication of the product rule. A check using a small systematic list confirms whether order is intended.
Key notes
Important points to keep in mind