# Concept derivation guides

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Concept guide



## [Factorial: uniqueness of the recursive definition, the product form and computing 5!](/en/derivations/derivation-mko-factorial-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-factorial)

Using only F(0)=1 and F(n+1)=(n+1)F(n) (n∈ℕ, with 0∈ℕ): (1) prove that at most one function ℕ→ℕ satisfies these two equations; (2) prove F(n)=∏_{k=1}^{n} k for every n (the empty product being 1), so that for n>0, n! is the product of all integers from 1 to n; (3) prove F(n)≥1 for every n, in particular 0!=1≠0; (4) compute 5!=120 step by step from the recursion.

Factorial

derivation-mko-factorial-01





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