Sum rule
If you are choosing one option from disjoint cases, add the counts.
a + b + c
Signal words: either, one of, separate cases.
Reference 1
Counting cheat sheet
The shortest reliable map for first-step combinatorics decisions.
Product rule
If a task is done in stages, multiply the number of choices per stage.
a · b · c
Signal words: first, then, for each.
Permutations
Use when you arrange distinct items and order matters.
n! or P(n,r) = n! / (n-r)!
Signal words: line up, arrange, rank, seat.
Combinations
Use when you choose items and order does not matter.
C(n,r) = n! / (r!(n-r)!)
Signal words: choose, select, committee, team.
Fast decision test
- Am I splitting into separate cases, or moving through stages?
- If I swap the chosen objects, do I get a different outcome?
- Am I counting valid outcomes directly, or should I count the total and subtract bad cases later?