A permutation is an arrangement of items in a specific order.
The number of ways to arrange n distinct items is n! (n factorial). The number of ways to choose and arrange k items from n items is P(n,k) = n!/(n-k)!. In permutations, order matters (unlike combinations). For example, arranging 3 books on a shelf from 5 books is a permutation: P(5,3) = 5×4×3 = 60. Permutations are used in probability, cryptography, scheduling, and counting problems.
Permutations were studied in ancient India for Sanskrit poetry and music. The 11th-century Indian mathematician Bhaskara II studied arrangements. The modern theory of permutations was developed by Augustin-Louis Cauchy and Joseph-Louis Lagrange in the 18th-19th centuries, leading to group theory. Permutations are fundamental in abstract algebra and combinatorics.