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Stars and bars — AMC & AIME Problems

Practice every real AMC 8, AMC 10, AMC 12, and AIME problem that uses stars and bars — 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.

All of these problems are used with official legal permission of the Mathematical Association of America (MAA).

AMC 8

2004 AMC 8 Problem 17 · 2018 AMC 8 Problem 23 · 2019 AMC 8 Problem 25

AMC 10

2001 AMC 10 Problem 19 · 2003 AMC 10A Problem 21 · 2016 AMC 10A Problem 20 · 2018 AMC 10A Problem 11 · 2025 AMC 10B Problem 22

AMC 12

2009 AMC 12B Problem 22 · 2010 AMC 12A Problem 25 · 2018 AMC 12B Problem 22

AIME

1998 AIME Problem 7 · 2000 AIME II Problem 5 · 2001 AIME I Problem 6 · 2004 AIME II Problem 8 · 2009 AIME I Problem 10 · 2011 AIME II Problem 7 · 2020 AIME I Problem 9 · 2021 AIME I Problem 4 · 2022 AIME II Problem 13 · 2024 AIME II Problem 3

More combinatorics concepts

arrangements with restrictionsbasic countingbijectionBurnside’s LemmaCatalan Numbercircular arrangementscombinationscombinatorial gamecomplementary countingcounting integers in a rangecounting intersectionscounting pairscounting regionscounting shapes in figuresderangementsdouble countingfencepost countinggenerating functionsgraph theoryinclusion-exclusionlattice pathsmultiplication principlemultiset permutationspartitions and compositionsPascal’s Trianglepermutationspigeonhole principlerecursive countingsubsetstilingtree diagramVandermonde’s ConvolutionVenn Diagram