What is 8 factorial ?

In mathematics, a factorial is a function applied to natural numbers that results in the product of an integer and all the positive integers that come before it. The factorial of 8 is significant because it not only represents a simple multiplication of numbers from 1 to 8 but also has numerous applications that are fundamental to various mathematical concepts, including combinatorics, probability, and sequence operations.

The formula for calculating a factorial is relatively straightforward:

n! = n × (n-1) × … × 1

For the factorial of 8, you would calculate:

8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1

By following these steps, you end up with:

8! = 40320

The factorial of 8 has several applications in different fields:

• Combinatorics: Used in counting permutations where order matters. For instance, the number of different ways to arrange 8 distinct books on a shelf is calculated as 8!.
• Probability Theory: In calculating probabilities where ordering is integral to the outcome of an event.
• Algebra: The factorial function appears in equations and identities, including the famous Binomial Theorem.

Test your understanding of factorials with these simple problems:

1. If a code has 8 distinct symbols, how many different 8-symbol codes can you create?
2. There are 8 runners in a race. How many different ways can the runners finish first, second, and third?

Here are the solutions:

1. The total number of different 8-symbol codes is the factorial of 8, which is 8! = 40320.
2. For the runners, the problem is a permutation of 8 items taken 3 at a time. This gives us 8P3 = 8! / (8-3)! = 8 × 7 × 6 = 336 different ways.