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The value 0! = 1 is not merely a convention — it follows naturally from the recursive definition of the factorial function and from combinatorics. The most direct justification is the recurrence relation: n! = n × (n−1)!. Setting n = 1 gives 1! = 1 × 0!, and since 1! = 1, it follows that 0! = 1. A combinatorial interpretation reinforces this: n! counts the number of ways to arrange n distinct objects in a sequence (permutations). There is exactly one way to arrange zero objects — the empty arrangement — so 0! = 1. A third justification comes from the Gamma function, an extension of the factorial to real and complex numbers: Γ(n) = (n−1)! for positive integers. Evaluating Γ(1) = ∫₀^∞ e^(−t) dt = 1, so (1−1)! = 0! = 1. All three perspectives — the recurrence relation, combinatorics, and the Gamma function — converge on the same answer.
answered by Omniscientia Team · 163 words · 18 Mar 2026