Textbook Question
A __________ is an ordered arrangement of r objects chosen from n distinct objects without repetition.
60
views
4. Probability
Counting
1:15 minutesProblem 5.RE.12b
Textbook Question
"Determine the value of each of the following:
b. 0!"
Verified step by step guidance1
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., \(n! = n \times (n-1) \times \cdots \times 2 \times 1\).
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.
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).
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.
Summarize that \$0! = 1$ by definition, which is a fundamental fact used throughout statistics and mathematics.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
Recommended video:
07:11
Introduction to Permutations
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.
Recommended video:
Guided course
02:59The Binomial Experiment Example 2
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.
Recommended video:
07:10Empirical Rule of Standard Deviation and Range Rule of Thumb
Related Videos
Related Practice