"Determine the value of each of the following:


b. 0!"

Verified step by step guidance

1. Recall the definition of the factorial function: for any positive integer \(n\), the factorial \(n!\) is the product of all positive integers from 1 up to \(n\), i.e., \(n! = n \times (n-1) \times \cdots \times 2 \times 1\).
2. Understand that the factorial of zero, \$0!$, is a special case and is defined to be 1 by convention. This is important for consistency in many mathematical formulas, especially in combinatorics and series expansions.
3. Recognize that this definition ensures that formulas involving factorials work smoothly even when \(n=0\), such as the number of ways to choose zero items from a set, which is exactly one way (choosing nothing).
4. Therefore, the value of \$0!$ is defined as 1 without performing any multiplication, as there are no positive integers less than or equal to zero to multiply.
5. Summarize that \$0! = 1$ by definition, which is a fundamental fact used throughout statistics and mathematics.

Key Concepts

Factorial Function

The factorial of a non-negative integer n, denoted n!, is the product of all positive integers less than or equal to n. It is commonly used in permutations, combinations, and probability calculations.

Definition of 0!

By convention, the factorial of zero (0!) is defined as 1. This definition ensures consistency in mathematical formulas, especially in combinatorics, where 0! represents the number of ways to arrange zero objects.

Combinatorial Interpretation of 0!

0! = 1 can be understood as the number of ways to arrange an empty set, which is exactly one way—the empty arrangement. This interpretation helps justify why 0! is set to 1 in counting problems.