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**.
The 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
Tiny 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 this kind of problem
- 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.
Next recommended lesson
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