What is 5 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.

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.

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?

• 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.