What is 5 factorial ?

Steps to calculate factorial of 5

To find 5 factorial, or 5!, simply use the formula that multiplies the number 5 by all positive whole numbers less than it.

Let’s look at how to calculate the Factorial of 5:

5! is exactly :
Factorial of 5 can be calculated as:
5! = 5 x 4 x 3 x 2 x 1

Factorials of Numbers similar to 5

What is Factorial?

The concept of factorial is used widely in mathematics to describe the product of a sequence of natural numbers descending from a specified number down to 1. Specifically, the factorial of 5, denoted as 5!, is significant because it provides the number of different ways in which a set of 5 distinct items can be arranged. Understanding the factorial is essential in disciplines that require counting arrangements or permutations, such as combinatorics and probability.

Formula to Calculate the Factorial of 5

The calculation of a factorial can be done using the formula n! = n × (n-1) × … × 1. Applying this formula to the number 5, we can determine 5 factorial as follows:

  • Step 1: Start with the number 5.
  • Step 2: Multiply 5 by one less than itself, which is 4, resulting in 20 (5 × 4).
  • Step 3: Continue this process with the next descending integers: 20 × 3 = 60, then 60 × 2 = 120.
  • Step 4: Finally, multiply the result by 1 (although this does not change the product): 120 × 1 = 120.

Hence, 5! equals 120.

What is the Factorial of 5 Used For?

Factorial of 5, or 5!, has several interesting applications:

  • In combinatorics, 5! is the number of ways you can arrange 5 different books on a shelf.
  • In probability theory, if there are 5 different races in a relay match, 5! represents the different ways the races can be ordered.
  • Furthermore, 5! is used in calculating combinations and permutations in statistics and helps in solving various problems related to the organization of data.


  • If a password is made by arranging 5 distinct letters, how many different passwords can be formed?
  • If you are to choose the first 5 leading runners in a race, how many different arrangements can be made?

Solutions to Exercises

  • Since there are 5 distinct letters and each can be used only once, the number of different passwords is equal to 5 factorial, which is 5! = 120.
  • There are 5! ways to arrange the order of the first 5 leading runners, meaning there are 120 distinct arrangements.

Frequently Asked Questions

Q: Why is the value of 0 factorial defined as 1?

A: The value of 0 factorial is 1 due to the convention of the nullary product – the result of multiplying no factors. In terms of combinatorics, there is exactly one way to arrange zero items, which is to arrange nothing, hence 0! = 1.

Q: Can the factorial function be applied to negative numbers?

A: No, the factorial function is only defined for non-negative integers. Negative numbers do not have a meaningful factorial result because a factorial represents a count of permutations, which cannot be negative.

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