COUNTING & PROBABILITY
Combinatorics, permutations, probability, expected value, and the pigeonhole principle.
What Counting & Probability mean on the AMC10
This section is about counting carefully without double-counting and translating word problems into probability expressions. Key tools:
- Counting principles — addition, multiplication, complementary counting
- Permutations and combinations — and
- The pigeonhole principle — "with pigeonholes and pigeons..."
- Probability basics — favorable / total, independence, conditional
- Expected value — weighted averages of outcomes
- Geometric probability — area- or length-based probability
When stuck, try complementary counting ("what's the probability it does not happen") — it cuts many problems in half.
SUBTOPICS
The foundational rules — multiplication, addition, complementary counting, casework, and inclusion-exclusion — that let you count complex sets quickly and accurately.
The probability-weighted average of all outcomes; linearity of expectation turns hard problems into sums of simple indicator variables.
Probability computed as a ratio of lengths, areas, or volumes — turning "pick a random point" questions into geometry problems.
Ordered and unordered counting, circular arrangements, repeated elements, and distributing identical objects.
If more objects than containers exist, at least one container must hold more than one object — a simple idea with powerful consequences.
Probability as favorable outcomes over total outcomes, complementary counting, independent and dependent events, conditional probability, and probability with combinatorics.