Counting principles
105 problems · Sum rule, product rule, bijections and double counting.
Ex.185Four-digit numbers with distinct digits
Counting principles
High school
Ex.226Counting functions
Counting principles
High school
Ex.227Handshakes at a party
Counting principles
High school
Ex.228The pigeonhole principle
Counting principles
High school
Ex.229The product rule
Counting principles
High school
Ex.230The sum rule
Counting principles
High school
Ex.322Arrangements of MISSISSIPPI
Counting principles
High school
Ex.335Stars and bars
Counting principles
L1
Ex.437Arrangements
Counting principles
High school
Ex.438The binomial theorem
Counting principles
High school
Ex.439Pascal's identity
Counting principles
High school
Ex.440Expanding $(x-y)^n$
Counting principles
High school
Ex.441Choosing with repetition
Counting principles
High school
Ex.442Combinations
Counting principles
High school
Ex.443Choosing a mixed committee
Counting principles
High school
Ex.444Counting injections
Counting principles
High school
Ex.445Permutations
Counting principles
High school
Ex.446Pigeonhole with balls
Counting principles
High school
Ex.447Subsets and their complements
Counting principles
High school
Ex.448Round-robin tournament
Counting principles
High school
Ex.602Dominoes on a mutilated chessboard
Counting principles
M1
Ex.603Replacing two numbers by their difference
Counting principles
M1
Ex.604Regions cut by n lines
Counting principles
M1
Ex.607Two take-away games
Counting principles
M1
Ex.608The expected number of fixed points of a random permutation
Counting principles
M1
Ex.635A subtraction game with moves 1, 3 and 4
Counting principles
M1
Ex.638Tiling a 2 × n board with dominoes and squares
Counting principles
M1
Ex.871Circular arrangements
Counting principles
L1
Ex.872Arrangements with repetition
Counting principles
High school
Ex.873A bijective proof
Counting principles
Olympiad
Ex.874Ordering books on a shelf
Counting principles
High school
Ex.875Counting Boolean functions
Counting principles
L1
Ex.876Counting cliques
Counting principles
Olympiad
Ex.877Coloring problems
Counting principles
L1
Ex.878Combinations with repetition
Counting principles
High school
Ex.879Complement counting
Counting principles
High school
Ex.880Degrees in a graph
Counting principles
L1
Ex.881Handshake lemma
Counting principles
L1
Ex.882Counting pairs within subsets
Counting principles
Olympiad
Ex.883Counting with at least one
Counting principles
L1
Ex.884At least one property
Counting principles
Olympiad
Ex.885Counting unions
Counting principles
L1
Ex.886Counting configurations
Counting principles
L1
Ex.887Sets of even size
Counting principles
Olympiad
Ex.888Exact intersection count
Counting principles
Olympiad
Ex.889Counting onto functions
Counting principles
Olympiad
Ex.890Monotone lattice paths
Counting principles
Olympiad
Ex.891Pigeonhole with integers
Counting principles
Olympiad
Ex.892Generalized pigeonhole
Counting principles
L1
Ex.893Pigeonhole proof
Counting principles
L1
Ex.894Subset sums
Counting principles
Olympiad
Ex.895Scheduling tournaments
Counting principles
L1
Ex.1234The game of coins on a round table
Counting principles
Olympiad
Ex.1237Finding the heavy coin with a balance
Counting principles
Olympiad
Ex.1238Flipping rows and columns of a board
Counting principles
Olympiad
Ex.1241A finite set in which every point is a midpoint
Counting principles
Olympiad
Ex.1243Subsets without two consecutive integers
Counting principles
Olympiad
Ex.1244Tiling a 10 × 10 board with 1 × 4 tiles
Counting principles
Olympiad
Ex.1246Tiling a 2ⁿ × 2ⁿ board with L-trominoes
Counting principles
Olympiad
Ex.1247Binary strings with no three equal consecutive digits
Counting principles
M1
Ex.1250Gray codes: a cyclic list of all bit strings with one change at a time
Counting principles
Olympiad
Ex.1253The expected number of records
Counting principles
M1
Ex.1254Sorting by adjacent swaps needs exactly the number of inversions
Counting principles
M1
Ex.1257Gas stations on a circular road
Counting principles
Olympiad
Ex.1307A 7 × 7 grid with no rectangle of marked cells
Counting principles
Olympiad
Ex.1309Partitions into distinct parts and into odd parts
Counting principles
Olympiad
Ex.1312Nim and the XOR rule
Counting principles
M1
Ex.1313The 15-puzzle with two tiles swapped
Counting principles
Olympiad
Ex.1603Generating function for binomial coefficients
Counting principles
Olympiad
Ex.1604Catalan applications
Counting principles
Olympiad
Ex.1605A combinatorial identity
Counting principles
L1
Ex.1606Derangements
Counting principles
L1
Ex.1607Double counting
Counting principles
Olympiad
Ex.1608Another double count
Counting principles
Olympiad
Ex.1609Alternating sum identity
Counting principles
Olympiad
Ex.1610Counting coprime integers
Counting principles
L1
Ex.1611Derangements via inclusion-exclusion
Counting principles
Olympiad
Ex.1612Derangement formula
Counting principles
Olympiad
Ex.1613A combinatorial identity via inclusion-exclusion
Counting principles
Olympiad
Ex.1614Surjection formula proof
Counting principles
Olympiad
Ex.1615Sieve formula
Counting principles
Olympiad
Ex.1616Surjections via inclusion-exclusion
Counting principles
Olympiad
Ex.1617Surjections onto $3$ elements
Counting principles
Olympiad
Ex.1618Three-set inclusion-exclusion
Counting principles
L1
Ex.1619Three-set counts
Counting principles
L1
Ex.1620Counting with matrices
Counting principles
Olympiad
Ex.1621The multinomial theorem
Counting principles
L1
Ex.1622Lattice paths
Counting principles
Olympiad
Ex.1623Permutations with fixed points
Counting principles
L1
Ex.1624Pigeonhole with 100 numbers
Counting principles
Olympiad
Ex.1625Multiples via pigeonhole
Counting principles
Olympiad
Ex.1626Pigeonhole in a graph
Counting principles
Olympiad
Ex.1627Pigeonhole with points
Counting principles
Olympiad
Ex.1628Sum via pigeonhole
Counting principles
Olympiad
Ex.1629Counting surjections
Counting principles
L1
Ex.1630Vandermonde's identity
Counting principles
Olympiad
Ex.1942The ballot problem
Counting principles
Olympiad
Ex.1972Chomp and the strategy-stealing argument
Counting principles
Olympiad
Ex.1975Oddtown
Counting principles
Olympiad
Ex.2116Bertrand's ballot problem
Counting principles
Olympiad
Ex.2117Catalan numbers
Counting principles
Olympiad
Ex.2226Subsets whose sum is divisible by p
Counting principles
Olympiad
Ex.2235Every set of n integers contains a sum-free subset of more than n/3 elements
Counting principles
Olympiad
Ex.2236Conway's soldiers cannot reach the fifth row
Counting principles
Olympiad
Ex.2237The Erdős–Ko–Rado theorem
Counting principles
Olympiad
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