AMC 10 Counting & Probability — Paper 2
Premium AMC 10 counting practice paper: 30 hard problems on expectation, conditional probability, inclusion-exclusion and stars and bars, with solutions.
About this test
AMC 10 Counting & Probability — Paper 2 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
- Three dice — 1 question
- Committees with roles — 1 question
- Runs and adjacency — 1 question
- Expected payoff — 1 question
- Coin combinations — 1 question
- Conditional on parity — 1 question
- Block arrangements — 1 question
- Random walks — 1 question
- Triangles from points — 1 question
- Card probability — 1 question
- Onto distributions — 1 question
- Expected maximum — 1 question
- Digit sums — 1 question
- Pigeonhole probability — 1 question
- Paths with an obstacle — 1 question
- Adjacency probability — 1 question
- Counting rectangles — 1 question
- Majority outcomes — 1 question
- Injective functions — 1 question
- Differences — 1 question
- Multinomial splits — 1 question
- Weighted total probability — 1 question
- Palindromes — 1 question
- At least one success — 1 question
- Non-decreasing sequences — 1 question
- With replacement — 1 question
- Divisor pairs — 1 question
- Expected fixed points — 1 question
- Bounded subsets — 1 question
- Triangle formation — 1 question
Difficulty spread
- 7 medium 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
How does this compare with Paper 1?
It is the premium tier. Paper 1 asks you to apply one counting idea per question; this one combines two, such as inclusion-exclusion inside an expectation, or a pigeonhole argument that makes a probability equal to 1.
Which techniques appear?
Complementary counting, inclusion-exclusion, stars and bars, linearity of expectation, conditional probability, onto distributions, multinomial splits, random walks, and counting paths around an obstacle.
How were the answers checked?
Almost every answer here was produced by enumerating the entire sample space in code rather than by applying a formula, then confirmed a second time by an independent derivation. The written solution gives the clean argument a contestant would use.
How is it scored?
Real AMC 10 rules: 6 points for a correct answer, 1.5 for a blank, 0 for a wrong one. At this difficulty a blank genuinely beats a guess unless you have ruled out three options.