Counting / Product And Sum Rules
Least You Need to Know: Product Rule and Sum Rule
Many counting errors come from not noticing whether choices happen in sequence or belong to separate non-overlapping cases.
Least you need to know
- Use the product rule for a sequence of independent choices.
- Use the sum rule for separate cases that do not overlap.
- If cases overlap, direct addition may overcount.
- Write down the stages of the choice before computing.
Key notation
- m × n — product rule
- m + n — sum rule
- |A| — size of a set
- A ∪ B — union of sets
Worked example
- Example: 3 shirt choices and 4 pants choices give 3 × 4 = 12 outfits.\n- This is a sequence of choices, so multiply.\n- If instead you choose a red hat or a blue hat from separate shelves with no overlap, you add.
Common mistakes
- Students often add when they should multiply.
- Students often multiply overlapping cases.
- Students often forget to define the stages of a choice.
How to recognize it
- Ask: am I making one choice after another?
- Ask: are these cases disjoint?
- If the same outcome can appear in two cases, watch for overcounting.
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