AMC 10 Permutations & Combinatorics — Paper 2
Premium AMC 10 permutations paper: 30 hard problems on derangements, Catalan paths, Burnside, surjections and bounded stars and bars, with solutions.
About this test
AMC 10 Permutations & Combinatorics — 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
- Fixed relative order — 1 question
- Triple inclusion-exclusion — 1 question
- Derangements of five — 1 question
- Multisets — 1 question
- Circular with restrictions — 1 question
- Weighted binomial sums — 1 question
- Catalan paths — 1 question
- Surjections — 1 question
- Strings avoiding a pattern — 1 question
- Pigeonhole principle — 1 question
- Committees with two roles — 1 question
- Vandermonde's identity — 1 question
- Non-adjacent repeats — 1 question
- Bounded stars and bars — 1 question
- Rotational equivalence — 1 question
- Expected minimum — 1 question
- Triangles in a grid — 1 question
- Identical boxes — 1 question
- At least one special — 1 question
- Staircase recursions — 1 question
- Rectangles excluding squares — 1 question
- Exactly one fixed point — 1 question
- Majority strings — 1 question
- Unlabelled teams — 1 question
- Monotone functions — 1 question
- Coprime counting — 1 question
- Exactly three fixed — 1 question
- Lattice points under a line — 1 question
- Ordered compositions — 1 question
- Subset sums — 1 question
Difficulty spread
- 1 medium question
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 Combinatorics 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 which of permutation or combination applies; this one assumes you know and asks you to combine that with inclusion-exclusion, a recursion, or a symmetry argument.
Which techniques appear?
Derangements, Catalan paths, Burnside's lemma for rotational symmetry, surjection counting, bounded stars and bars, Vandermonde's identity, the pigeonhole principle, Stirling numbers for identical boxes, and Fibonacci recursions.
How were the answers checked?
Nearly every one was produced by enumerating the whole sample space in code, then re-derived a second time independently before the paper was published. A script answering with the stored keys scores full marks.
How is it scored?
Real AMC 10 rules: 6 points for a correct answer, 1.5 for a blank, 0 for a wrong one, over 75 minutes.