Combinatorics
Combinatorics is the branch of mathematics that studies counting, arranging, selecting, and analyzing discrete objects and structures.
Set: A collection of distinct objects, treated as a single mathematical entity. Combinatorial problems often begin by specifying the objects available for selection.
Product rule: A counting rule that multiplies the numbers of choices made in successive independent stages. It counts arrangements by splitting their construction into sequential choices.
Graph theory: The mathematical study of graphs, consisting of vertices joined by edges. Graphs are discrete structures whose paths, matchings, and colorings are counted and analyzed.
Calculus: The branch of mathematics concerned with limits, derivatives, integrals, and continuous change. Its central tools address continuous variation, unlike combinatorics’ discrete structures.
Blaise Pascal: A seventeenth-century French mathematician whose work helped establish probability theory and projective geometry. His study of the arithmetic triangle shaped early techniques for counting selections.
Permutation: An ordered arrangement of objects, usually drawn from a specified set. Permutations count arrangements in which changing the order creates a different outcome.
Sum rule: A counting rule that adds the sizes of mutually exclusive alternatives. It handles cases where an outcome belongs to exactly one of several classes.
Design theory: The study of arrangements of subsets satisfying specified balance and intersection conditions. Its designs organize finite collections for experiments, communication, and error control.
Number theory: The branch of mathematics that studies integers and their properties. It overlaps with combinatorics in divisibility, partitions, and arithmetic counting problems.
Jacob Bernoulli: A Swiss mathematician whose work in probability and analysis was published posthumously in Ars Conjectandi. Ars Conjectandi systematized combinatorial methods and connected them with probability.