Counting with the Product and Sum Rules
Count a finite set by breaking the choice into independent stages to multiply, or into disjoint cases to add, and say which structure the problem has.
What a learner can do afterwards
- Decide whether a counting problem splits into stages or into cases, and justify the split
- Count arrangements where a later stage's options depend on an earlier choice
- Detect and repair a double count caused by treating overlapping cases as disjoint
1 · Read
Some counting jobs run in stages, and some split into cases. When you make one pick and then another, the stages multiply. When you choose one path among separate groups, the cases add. First name the structure, then compute.
Priya owns 3 shirts and 4 trousers and wears one of each. Each shirt pairs with each trousers, so she has 3 times 4, or 12 outfits. Five runners contest gold, silver, and bronze with no ties. Gold has 5 options, then silver has 4, then bronze has 3, so there are 5 times 4 times 3, or 60 outcomes.
Cases only add when they share no outcomes. Two soups and 3 salads as one starter give 2 plus 3, or 5 choices, since no dish is both. When cases overlap, adding double counts the shared outcomes, so subtract them once: 10 French students plus 8 German students minus 3 who take both gives 15.
Before you compute, ask whether you are doing one thing after another or choosing one path. Stages multiply even when later options depend on earlier picks. Cases add only when no outcome sits in two cases at once.
Multiply the stages of a sequence, add the cases of a split, and subtract any overlap.
More to read
7.1 The Multiplication Rule for Counting
The multiplication rule counts choices made in stages. If stage one has m options and stage two has n options for each first choice, the total is m times n. Outfits, menus with courses, and PIN codes all work this way. The rule extends to any number of stages. Always check the later count does not secretly change before you multiply.
7.2 Permutations
Permutations count ordered arrangements, where later options depend on earlier picks. Awarding medals to runners gives 5, then 4, then 3 options, and you still multiply the actual counts. This is the product rule with dependent stages. The justification is the same: every first choice branches into the same number of continuations. Ordered outcomes are the natural home of multiplication.
13.5 Counting Principles
Some problems split into cases instead of stages, and then you add. Choosing one starter from soups or salads is the classic shape. The cases must be disjoint, with no outcome sitting in two cases at once. When cases overlap, adding double counts the shared outcomes, and you must repair the count. So the first job is always to name the structure: stages to multiply, or disjoint cases to add.
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Where it sits
14 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.