Why Does 0! Equal 1 Rather Than 0?

A factorial can be understood as a count of arrangements: three distinct objects have 3 × 2 × 1 = 6 possible orders. With no objects, it may seem that there are “no arrangements”, but that confuses having no objects with having no outcome. The empty set still has one arrangement: place nothing and do nothing. There is no second, different order. Hence 0! = 1.

The same conclusion follows from the factorial recurrence. For a positive integer n, n! = n × (n − 1)!. Setting n to 1 gives 1! = 1 × 0!. Since 1! = 1, zero factorial must also equal 1 if the rule is to remain valid at the boundary. This is also an empty product: a product with no factors takes the value 1 because 1 is the multiplicative identity and does not alter any factor later included.

So 0! = 1 does not claim that “zero things” are one thing, and it has nothing to do with multiplying by zero. It counts possible arrangements, not objects, and the empty arrangement is exactly one outcome. This boundary value keeps combinatorial formulas coherent. For example, there is exactly one way to choose nobody from five people: choose nobody.

https://dlmf.nist.gov/5.2