combinatorics
Arrangements with restrictions — AMC & AIME Problems
77 problems
Practice every real AMC 8, AMC 10, AMC 12, and AIME problem that uses Arrangements with restrictions — a combinatorics concept — each with a professionally curated solution, many with a video solution too, from LIVE by Po-Shen Loh. Browse all concepts or the full past-contest archive.
LIVE focuses on challenge. As a social enterprise, we also explain standard school concepts one at a time, for free, on our sister site Expii. For this idea: What Is a Permutation?, What Is the Fundamental Counting Principle?.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
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These problems’ difficulty ratings come from real solve attempts on our platform. Tap a band or a contest to filter the list below, and tap a selected band again to include all difficulties. See the full difficulty guide.
AMC 8
AMC 10
- 2000 AMC 10 Problem 13
- 2006 AMC 10A Problem 18
- 2007 AMC 10B Problem 20
- 2008 AMC 10B Problem 21
- 2008 AMC 10B Problem 22
- 2010 AMC 10B Problem 23
- 2012 AMC 10B Problem 11
- 2012 AMC 10B Problem 22
- 2013 AMC 10A Problem 24
- 2014 AMC 10A Problem 4
- 2015 AMC 10A Problem 10
- 2015 AMC 10A Problem 22
- 2017 AMC 10A Problem 19
- 2018 AMC 10A Problem 4
- 2019 AMC 10B Problem 25
- 2021 AMC 10A Spring Problem 20
- 2021 AMC 10A Spring Problem 25
- 2021 AMC 10A Fall Problem 18
- 2021 AMC 10A Fall Problem 24
- 2022 AMC 10A Problem 9
- 2022 AMC 10A Problem 14
- 2022 AMC 10A Problem 24
- 2023 AMC 10A Problem 20
- 2024 AMC 10A Problem 20
- 2024 AMC 10B Problem 20
AMC 12
AIME
- 1984 AIME Problem 11
- 1986 AIME Problem 13
- 1988 AIME Problem 15
- 1990 AIME Problem 8
- 1997 AIME Problem 8
- 2001 AIME I Problem 14
- 2003 AIME II Problem 3
- 2004 AIME I Problem 6
- 2005 AIME I Problem 5
- 2006 AIME II Problem 4
- 2007 AIME I Problem 10
- 2008 AIME I Problem 11
- 2008 AIME II Problem 12
- 2009 AIME I Problem 10
- 2011 AIME I Problem 12
- 2011 AIME II Problem 7
- 2012 AIME II Problem 3
- 2015 AIME II Problem 12
- 2016 AIME II Problem 13
- 2018 AIME I Problem 3
- 2019 AIME II Problem 5
- 2020 AIME I Problem 5
- 2020 AIME II Problem 9
- 2023 AIME II Problem 7
- 2024 AIME II Problem 9
- 2025 AIME I Problem 7
- 2025 AIME I Problem 10
- 2025 AIME II Problem 3
- 2025 AIME II Problem 10
More combinatorics concepts
- Basic counting
- Bijection
- Burnside’s Lemma
- Catalan Number
- Circular arrangements
- Combinations
- Combinatorial game
- Complementary counting
- Counting integers in a range
- Counting intersections
- Counting pairs
- Counting regions
- Counting shapes in figures
- Derangements
- Double counting
- Fencepost counting
- Generating functions
- Graph theory
- Inclusion-exclusion
- Lattice paths
- Multiplication principle
- Multiset permutations
- Partitions and compositions
- Pascal’s Triangle
- Permutations
- Pigeonhole principle
- Recursive counting
- Stars and bars
- Subsets
- Tiling
- Tree diagram
- Vandermonde’s Convolution
- Venn Diagram