AMC 10 Counting & Probability — Paper 1
AMC 10 counting and probability practice paper: 30 problems on combinations, expected value, conditional probability and stars and bars, with full solutions.
About this test
AMC 10 Counting & Probability — Paper 1 is a full 30-question paper you attempt online in 75 minutes. It follows the American Mathematics Competition 10 pattern, so the timing, the question style and the marking match the real exam. There is no negative marking, so it is worth attempting every question. Leaving a question blank still earns 1.5 marks.
You get your score the moment you submit, question by question, with a step-by-step solution to every problem. 25 questions in 75 minutes. A high score qualifies for the AIME.
What it covers
Sections
- Dice probability — 1 question
- Combinations — 1 question
- Arrangements — 1 question
- Complementary counting — 1 question
- Without replacement — 1 question
- Lattice paths — 1 question
- Circular permutations — 1 question
- Expected value — 1 question
- Inclusion-exclusion — 1 question
- Distinct choices — 1 question
- Geometric probability — 1 question
- Conditional probability — 1 question
- Binomial counting — 1 question
- At least one — 1 question
- Derangements — 1 question
- Parity — 1 question
- Non-adjacent subsets — 1 question
- Order statistics — 1 question
- Stars and bars — 1 question
- Expected count — 1 question
- Adjacency restrictions — 1 question
- Ordering probability — 1 question
- Law of total probability — 1 question
- Digit counting — 1 question
- Collision probability — 1 question
- Dice sums — 1 question
- Counting divisors — 1 question
- Uniform spinners — 1 question
- Handshake counting — 1 question
- Subset parity — 1 question
Difficulty spread
- 4 easy questions
- 13 medium questions
- 2 hard questions
Before you start
15 questions · 45 minutes.
- Each correct answer scores 6 points.
- Each unanswered question scores 1.5 points.
- A wrong answer scores 0 — a blank is worth more than a guess unless you can eliminate at least three options.
- No calculators. Rough work on paper.
This paper covers Counting & Probability only, so a weak score points at one topic rather than leaving you guessing.
A look at the questions
Questions people ask
What does this paper cover?
Combinations and permutations, arrangements with repeated letters, complementary counting, conditional probability, expected value, geometric probability, derangements, stars and bars, and counting subsets with restrictions.
Why do so many solutions use the complement?
Because "at least one" almost always has fewer cases when you count what you do not want. Recognising when to flip to the complement is the single highest-value habit in AMC counting questions.
How is it scored?
Real AMC 10 rules: 6 points for a correct answer, 1.5 for a blank and 0 for a wrong one. A blank is worth more than a guess whenever you have not eliminated at least three options.
Are the answers checked?
Yes, and unusually thoroughly: almost every answer on this paper was produced by enumerating the entire sample space in code, not by applying a formula. The written solution then gives the clean argument.