AMC 10 Permutations & Combinatorics — Paper 1
AMC 10 permutations practice paper: 30 problems on arrangements, combinations, derangements, stars and bars and tilings, each with a full solution.
About this test
AMC 10 Permutations & Combinatorics — 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
- Basic permutations — 1 question
- Ordered selections — 1 question
- Repeated letters — 1 question
- Basic combinations — 1 question
- Circular seating — 1 question
- Sequential choices — 1 question
- Counting with a condition — 1 question
- Counting subsets — 1 question
- Diagonals and pairs — 1 question
- Forbidden positions — 1 question
- Alternating arrangements — 1 question
- Circular non-adjacency — 1 question
- Central binomial — 1 question
- Binary strings — 1 question
- Pascal's identity — 1 question
- Non-empty selections — 1 question
- Stars and bars with a floor — 1 question
- Two-stage paths — 1 question
- Derangements — 1 question
- Mixed selections — 1 question
- Restricted handshakes — 1 question
- Increasing digits — 1 question
- Tilings — 1 question
- Identical objects — 1 question
- Binomial sums — 1 question
- Adjacent-repeat strings — 1 question
- Triangles on a circle — 1 question
- Unordered pairs of pairs — 1 question
- Square divisors — 1 question
- Partial permutations — 1 question
Difficulty spread
- 3 easy questions
- 13 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 Combinatorics only, so a weak score points at one topic rather than leaving you guessing.
A look at the questions
Questions people ask
What is the difference between a permutation and a combination?
A permutation counts arrangements, where order matters; a combination counts selections, where it does not. Deciding which one a question wants is the single most common source of error, so several questions here pair the two counts so you can see them side by side.
What does this paper cover?
Arrangements with repeated letters, circular seating, combinations, stars and bars, derangements, Pascal's identity, tilings with a Fibonacci recursion, lattice paths, and counting with forbidden positions.
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, over 75 minutes.
How were the answers verified?
Wherever the sample space was small enough, the answer was produced by enumerating every case in code rather than by applying a formula, then confirmed a second time independently.