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

Filter by difficulty

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.

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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

1958 AMC 12 Problem 49 • 2009 AMC 12B Problem 22 • 2010 AMC 12A Problem 25 • 2018 AMC 12B Problem 22

AIME

1986 AIME Problem 13 • 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