Reference

Core counting ideas

A compact reference for the first combinatorics lessons.

1. Counting as choices

If a task happens in stages, count how many choices are available at each stage.

2. Multiplication principle

If step A has a choices and step B has b choices after that, then the total number of outcomes is a × b.

Example: 3 shirts and 4 trousers gives 12 outfits.

3. Addition principle

If you can do one of several disjoint things, add the counts.

Example: 5 math books or 2 physics books gives 7 possible books to pick.

4. Order matters vs order does not matter

• Order matters: arrangements, rankings, passwords.
• Order does not matter: committees, teams, chosen sets.

5. Fast self-check

• Am I building something in stages? Multiply.
• Am I choosing between separate cases? Add.
• Would swapping positions create a new outcome? If yes, order matters.