Ohio Assessments for Educators (OAE) Mathematics Practice Exam

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What is the factorial of 0 expressed as?

1
0
Undefined
Infinity

The correct answer is: 1

The factorial of 0 is expressed as 1, which may seem counterintuitive at first. The concept of a factorial involves multiplying a series of descending positive integers down to 1. For example, 5! (5 factorial) is calculated as 5 × 4 × 3 × 2 × 1 = 120. However, for 0!, the definition is established through combinatorial reasoning and the properties of factorials. The factorial function is defined such that n! = n × (n-1)! for any positive integer n. If we apply this definition to n = 1, we get: 1! = 1 × 0! Knowing that 1! equals 1, we can rearrange the equation to find 0!: 1 = 1 × 0! 0! = 1 / 1 0! = 1 This reasoning aligns with the concept of counting combinations, where the number of ways to choose 0 objects from a set of objects is exactly one way—by choosing nothing. Thus, defining 0! as 1 maintains consistency within mathematical frameworks, particularly in combinatorics and algebra.