Factorial

The factorial of a nonnegative integer n, written n!, is the product of all positive integers up to n; by convention, 0! = 1.

Factorial recurrence: The identity n! = n × (n − 1)!, with 0! = 1. It reduces each factorial to a smaller one and provides a direct computational rule.
Permutation: An ordered arrangement of distinct objects. The n! permutations of n distinct objects count every possible ordering.
Empty product: A product with no factors, defined as 1. The empty product convention explains why 0! equals 1.
Christian Kramp: A French mathematician who introduced the exclamation-mark notation for factorials in 1808. He established the notation now used to write n!.
Gamma function: A function that extends factorial values to most real and complex arguments, satisfying Γ(n) = (n − 1)! for positive integers. It extends the integer factorial to non-integer arguments.
Binomial coefficient: The number of ways to choose k objects from n, written as n choose k. Its factorial formula is n! divided by k!(n − k)!.
Product notation: A compact notation for multiplying a sequence of indexed terms. Factorial is a special case of a product over consecutive integers.
André-Louis Cholesky: A French military officer and mathematician associated with a matrix factorization named after him. The shared word can cause confusion: Cholesky factorization is unrelated to factorial products.
Prime factorization: The expression of an integer as a product of prime numbers. Counting prime factors in n! reveals its divisibility and supports exact computation.
Multinomial coefficient: A coefficient counting arrangements of objects divided into several specified groups. Factorial ratios count arrangements when repeated group members are indistinguishable.

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