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82 questions · full solutions

Combinatorics

Counting under pressure: paths, arrangements, derangements, Catalan structures, and the double-counting arguments that turn hard questions into two-line answers.

  1. 01HandshakesEasy
  2. 02Counting SubsetsEasy
  3. 03The Round TableEasy
  4. 04Grid WalksMedium · Private
  5. 05Arranging MISSISSIPPIMedium · Private
  6. 06Trailing Zeros of 100!Medium · Private
  7. 07Counting Two PairHard · Private
  8. 08No Adjacent OnesHard · Private
  9. 09Diagonals of a 12-gonEasy
  10. 10Committee With a QuotaEasy
  11. 11Climbing Ten StairsEasy
  12. 12Strictly Increasing DigitsEasy
  13. 13Ordered Sums of 10Easy
  14. 14Stars and BarsMedium · Private
  15. 15Paths Around a RoadblockMedium · Private
  16. 16Five Letters, Wrong EnvelopesMedium · Private
  17. 17Divisible by 2, 3, or 5Medium · Private
  18. 18Tiling a 2×12 StripMedium · Private
  19. 19Rectangles on a ChessboardMedium · Private
  20. 20Counting FlushesMedium · Private
  21. 21Counting StraightsMedium · Private
  22. 22Exactly One PairMedium · Private
  23. 23The Odd HandshakersMedium · Private
  24. 24Pairing Off Ten PeopleHard · Private
  25. 25A Weighted Binomial SumHard · Private
  26. 26Onto FunctionsHard · Private
  27. 27Balanced BracketsHard · Private
  28. 28Necklaces and BurnsideHard · Private
  29. 29No Couple TogetherHard · Private
  30. 30Two Subsets, Same SumHard · Private
  31. 31A Mixed Committee CountEasy
  32. 32No Three Zeros in a RowMedium · Private
  33. 33Capped Stars and BarsMedium · Private
  34. 34Triangulating a HeptagonMedium · Private
  35. 35The Ballot ProblemMedium · Private
  36. 36Making Change for Fifty CentsMedium · Private
  37. 37Exactly Two in PlaceHard · Private
  38. 38Lattice Paths Below the DiagonalHard · Private
  39. 39Tiling a 3-by-4 Board with L-TrominoesHard · Private
  40. 40One Must Divide AnotherEasy
  41. 41A PIN With a RepeatEasy
  42. 42Alternating in a RowEasy
  43. 43Exactly Four RunsMedium · Private
  44. 44Four Chairs, No NeighborsMedium · Private
  45. 45Three Trading PodsMedium · Private
  46. 46Every Way to Group FiveMedium · Private
  47. 47Three Different Parts of TwentyMedium · Private
  48. 48Paths With a Diagonal GearMedium · Private
  49. 49Quants Kept ApartMedium · Private
  50. 50Counting Ascents in a ShuffleMedium · Private
  51. 51Four Dice Hit EighteenMedium · Private
  52. 52The Unavoidable TrendMedium · Private
  53. 53Husbands in the GapsHard · Private
  54. 54Secret Santa, Couples EditionHard · Private
  55. 55No Venue Twice RunningHard · Private
  56. 56Twelve Lights in a RingHard · Private
  57. 57Dominoes on a 3-by-10 DeckHard · Private
  58. 58Dominoes and Squares TogetherHard · Private
  59. 59An Empty TillHard · Private
  60. 60Brackets That Stay ShallowHard · Private
  61. 61Subset Sums Mod ThreeHard · Private
  62. 62Distinct Parts, Odd PartsHard · Private
  63. 63Slicing Fifteen LotsHard · Private
  64. 64Three Cycles in a ShuffleHard · Private
  65. 65New Highs on the TapeHard · Private
  66. 66Paradoxical TriplesHard · Private
  67. 67Bracelets, Half BlackHard · Private
  68. 68Staffing Around a FeudHard · Private
  69. 69Two Wires, No ContactHard · Private
  70. 70A Streak Divisible by SevenHard · Private
  71. 71Seventeen Servers, Three ProtocolsHard · Private
  72. 72Seven Committees, One Overlap EachHard · Private
  73. 73Five Couples, Nobody Beside Their PartnerHard · Private
  74. 74A Three-Way Count That Never CrossesHard · Private
  75. 75Eight Servers, No Matching NeighborsHard · Private
  76. 76Wiring Seven ServersHard · Private
  77. 77Three Swaps from SortedHard · Private
  78. 78Ten Ticks in a CorridorHard · Private
  79. 79Two Desks per ProductHard · Private
  80. 80Flat Ticks AllowedHard · Private
  81. 81Paths That Skip the MiddleMedium · Private
  82. 82Prizes Around the CircleMedium · Private