Factorial

discrete-math

Definition

Factorial

The factorial of a non-negative integer is the product of all positive integers up to :

Equivalently, the factorial function is defined recursively by

Combinatorial Meaning

For a finite set with , the value is the number of permutations of its elements:

To construct a permutation, there are choices for its first position, for its second, and so on. The product rule therefore gives

Why Zero Factorial Is One

The empty set has exactly one permutation: the empty arrangement. Hence . This value is also forced by the recurrence,

which requires .

Examples

Arranging four objects

Four distinct objects can be ordered in

different ways.