Ohio Educators Math Exam Practice 2026 – Complete Prep Guide

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What is the factorial of 0 expressed as?

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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.

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