# Counting & Combinatorics

"How many ways are there to…?" sounds like a question you answer by listing them all and counting. For
anything real - possible passwords, lottery tickets, ways to seat a team - that list is astronomically
long, and listing is hopeless. Combinatorics is the art of getting the count *without* the list, and it
rests on a few small, powerful rules.

This guide builds them up: the multiplication principle (the engine behind almost all counting), then
the crucial fork between **permutations** (when order matters) and **combinations** (when it doesn't),
and finally why this matters far beyond puzzles - it's the foundation of probability, the reason a long
password is strong, and the reason some problems are too big to brute-force. By the end, "how many ways"
becomes a calculation, not a guess.

## How to read this
- **Want the one rule that does the most?** [Phase 1](01-the-multiplication-principle.md) - the
  multiplication principle.
- **Want the whole toolkit?** Read in order - permutations and combinations (Phase 2) build on it.

## The phases
1. **[The Multiplication Principle](01-the-multiplication-principle.md)** - counting independent choices
   by multiplying, and the "and vs or" rule.
2. **[Permutations & Combinations](02-permutations-and-combinations.md)** - order matters vs order
   doesn't, factorials, and the formulas (with runnable code).
3. **[Why Counting Matters](03-why-counting-matters.md)** - the bridge to probability, password
   strength, and combinatorial explosion (why brute force fails).

> This builds on [Numbers & Number Systems](/guides/numbers-and-number-systems) and the set idea from
> [Sets, Relations & Functions](/guides/sets-relations-and-functions). It sets up the last foundation:
> probability and statistics.