Why is 0 factorial defined as 1?

In mathematics, the definition of $0!=1$ is not merely an arbitrary convention, but a necessity for consistency across various mathematical formulas and functions.

While the standard definition of a factorial, $n!$, is the product of all positive integers from 1 to $n$, this definition does not naturally extend to zero. However, defining $0!$ as 1 allows for the seamless application of counting formulas in combinatorics. For example, the formula for the number of permutations of $n$ objects taken $k$ at a time is given by:12

$${}_n{P}_k=\frac{n!}{(n-k)!}$$

If we wish to calculate the number of ways to arrange all $n$ objects (where $k=n$), the formula becomes $n!/0!$. For this to equal $n!$, $0!$ must be defined as 1.3

Furthermore, this definition is supported by the gamma function, which generalizes the factorial to non-integral values. The gamma function satisfies the recursive property $\Gamma (x+1)=x\Gamma (x)$, and since $\Gamma (1)=1$, it follows that $0!$ corresponds to $\Gamma (1)$, maintaining mathematical consistency.4