factorial0

Factorial0 is a term used to describe the factorial of zero, commonly written as 0!. In standard factorial notation, for a nonnegative integer n, n! denotes the product 1 × 2 × ... × n. By convention, the case n = 0 yields an empty product, which is defined to be 1. This establishes a consistent base case for the factorial function and its recurrence relation.

The key justification for 0! = 1 lies in both algebra and combinatorics. Algebraically, the relation n! = n × (n−1)! for all n ≥ 1 requires a base value, and 0! = 1 satisfies this recurrence. Combinatorially, 0! counts the number of ways to arrange zero objects (there is exactly one empty arrangement) and appears in formulas such as binomial coefficients, where C(n, 0) = 1.

Connections to broader mathematics include the gamma function, which generalizes factorial to complex numbers: for nonnegative integers, n! = Γ(n+1), and thus 0! = Γ(1) = 1. In computer science and programming, the value 0! = 1 is used as the base case in recursive implementations of factorial and in algorithms that count arrangements or combinations.

In summary, factorial0 refers to the convention that the factorial of zero equals one, a convention that ensures consistency across algebraic identities, combinatorial interpretations, and analytic extensions.