Is 0 factorial infinity

0 factorial (0!) equals 1, not infinity or zero. This definition might seem counterintuitive—after all, how can multiplying "nothing" equal 1? However, the mathematical convention follows from the principle that factorial represents the number of ways to arrange items, and there is exactly one way to arrange zero items: the empty arrangement.

The definition 0! = 1 maintains consistency across mathematical formulas, particularly in combinatorics and calculus. For example, the binomial coefficient formula C(n,0) = n!/(0! × n!) correctly yields 1 for any n only when 0! = 1, reflecting the mathematical truth that there's one way to choose zero items from any set. Similarly, the exponential series e^x = Σ(x^n / n!) works correctly only when 0! = 1, as the first term must equal x^0 / 0! = 1/1 = 1. This convention also aligns with the factorial recurrence relation: since n! = n × (n-1)!, working backwards from 1! = 1 gives us 1! = 1 × 0!, therefore 0! must equal 1. Understanding this fundamental definition is crucial for working with permutations, combinations, probability distributions, and power series in mathematics and statistics.