Combinatorics
Count, compute odds and expect the unexpected, without listing every case.
- 1Permutations and CombinationsreadyLine things up, seat people at round tables, rearrange words, choose teams and count subsets: the counting toolkit behind every AMC combinatorics problem.
0% - 2ProbabilityreadyCount equally likely outcomes, decide whether order matters, multiply independent chances and follow probability trees when one event changes the next.
0% - 3CaseworkreadySplit a hard count into cases that don't overlap and miss nothing, then add them up: the casework habit, plus conditional probability.
0% - 4Principle of Inclusion and ExclusionreadyCount everything in A or B (or C, or more) without counting anything twice: add the sets, subtract the overlaps, add back, and pair it with complementary counting.
0% - 5Stars & BarsreadyShare identical objects among distinct bins by lining up stars and bars, then bend the trick to handle minimums, upper limits, parity, inequalities and seating gaps.
0% - 6Combinatorial IdentitiesreadyCount the same thing two ways and get identities for free: symmetry, row sums, Vandermonde, Pascal and the Hockey Stick, each one a shortcut through a sum that would take forever by hand.
0% - 7Geometric CountingreadyCount triangles, rectangles, squares, paths and polyhedron edges without listing them: choose the lines or points that define each shape, then let combinations and Euler's formula do the work.
0% - 8Geometric ProbabilityreadyWhen outcomes form a continuum, count them by measuring: lengths, areas and volumes turn waiting times, random points and dropped coins into probabilities.
0% - 9Expected ValuereadyWeighted averages, fair games and the most useful trick in contest probability: linearity of expectation with indicator variables.
0% - 10RecursionreadyCount something big by counting it small first: base cases, a recursion that builds each value from earlier ones, and two-state tables for harder rules.
0% - 11Probability StatesreadyTurn a game that could go on forever into a few equations: one unknown per state, symmetry to merge states, tables for step-by-step chances, and "+1" for expected values.
0%