combinare

Latin combinare meant to join together, to unite — from com (together) and bini (two together, in pairs). A combination was originally a joining, a coupling. In mathematics, a combination is a selection of items from a larger set where order does not matter — choosing 3 students from a class of 30 to form a committee, where the committee ABC is the same as BAC.

Pascal's Triangle, described by Blaise Pascal in 1653 (though known earlier in China and India), organizes combinatorial numbers visually: each number in the triangle is the sum of the two numbers above it, and each row gives the combinations for choosing 0, 1, 2, ... items from a set of that size. Pascal's Traité du Triangle Arithmétique was the first systematic treatment of combinatorics as a mathematical discipline.

Gottfried Leibniz's Dissertatio de Arte Combinatoria (1666) — published when he was 19 — proposed that all human knowledge could be reduced to combinations of fundamental concepts, and that a universal calculating machine could enumerate all possible combinations. This was among the earliest visions of computation as combinatorial process.

The combination lock takes its name from the mathematical concept — choosing the right combination from a set of possible sequences to open it. Cryptographic systems are essentially combination problems: how many ways can you choose keys of a given length from an alphabet of characters? The mathematics of combinations underlies the security of encrypted communication.