What is 3 factorial ?

A factorial is a mathematical operation designated by an exclamation point (!) used to define the product of an integer and all the natural numbers less than itself. For example, when looking at the factorial of 3, symbolized as 3!, we multiply 3 by every positive integer below it, which are 2 and 1. Understanding the factorial of 3 is significant because it lays the groundwork for advanced mathematical concepts such as permutations and combinations, essential in various fields, including statistics and number theory.

The general formula for calculating any factorial is n! = n × (n-1) × … × 1. To find the factorial of 3, we apply this formula:

• Start with the number itself, which is 3
• Multiply by the next smaller number, which is 2: 3 × 2
• Finally, multiply by the smallest positive integer, which is 1: 3 × 2 × 1

Following these steps, we arrive at the factorial of 3 being 3! = 6.

The factorial function has various applications especially in fields such as combinatorics and probability theory. Specifically, 3! is often used when determining the number of different ways to arrange or select 3 distinct items. Understanding the outcome of 3! can help solve fundamental problems in arranging objects or even in basic algorithms in computer science.

• Question 1: If 3 different books are to be arranged on a shelf, in how many different ways can this be done?
• Question 2: In a race where 3 friends compete, how many possible outcomes are there for the first, second, and third place?

Solution 1: The number of ways to arrange 3 books is equal to the factorial of 3, which is 3! = 6. Thus, there are 6 distinct arrangements.

Solution 2: The number of possible outcomes for them finishing the race is also 3!, which is 6 potential orders of finish.