How is 0! (zero factorial) equal to 1 (mathematical proof)?
The correct answer and explanation is:
The value of 0 factorial, written as 0!, is defined to be 1. This might seem unintuitive at first because factorial is the product of all positive integers up to a given number. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Since zero has no positive integers less than or equal to it, the definition of 0! requires a different reasoning.
One way to understand why 0! = 1 comes from the concept of the factorial function as the number of ways to arrange or permute objects. The factorial n! represents the number of ways to arrange n distinct objects. When n = 0, there is exactly one way to arrange nothing — the empty arrangement. Since the count of ways must be a number, it is defined as 1.
From a more mathematical point of view, the factorial function is closely related to the Gamma function. The Gamma function Γ(n) generalizes the factorial function to complex and real numbers. For positive integers n, Γ(n) = (n − 1)!. It is known that Γ(1) = 1. Thus, 0! = Γ(1) = 1.
Another rigorous way to justify 0! = 1 is through the recursive property of factorials:
n! = n × (n − 1)!
For n = 1, this becomes
1! = 1 × 0!
Since 1! = 1 by definition, the equation simplifies to
1 = 1 × 0!
Dividing both sides by 1 yields
0! = 1
This recursive definition ensures the factorial function behaves consistently for all integers starting from zero.
Additionally, the formula for combinations, which depends on factorials,
C(n, k) = n! / (k! (n − k)!)
must be valid when k = 0 or k = n. Since C(n, 0) counts the number of ways to choose zero items from n, it equals 1. To keep this true, 0! must be defined as 1.
In summary, defining 0! as 1 maintains consistency across mathematical definitions, recursive formulas, and combinatorial interpretations. This definition is universally accepted and forms a fundamental part of many mathematical formulas and proofs.