What is 9 factorial ?

Factorial, symbolized by an exclamation mark (!), is a mathematical concept that represents the product of all positive integers from 1 up to a given number. The factorial of 9, denoted as 9!, is particularly significant in mathematics because it reflects the number of ways you can arrange or permute 9 distinct items. Understanding factorials is fundamental in various areas of mathematics, including combinatorics and probability theory.

The basic formula to calculate the factorial of a number n is n! = n × (n-1) × … × 1. Applying this formula to the factorial of 9, we follow these steps:

• Start with the number 9.
• Multiply 9 by every natural number less than itself down to 1.
• 9! = 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1.

Following the calculation, the factorial of 9 equals 362,880.

Factorials serve as a key tool in several mathematical fields. The factorial of 9, or 9!, has its applications in areas such as:

• Combinatorics: Determining the number of possible combinations in a lottery or a card game.
• Probability Theory: Calculating the odds of certain events occurring within a fixed set of outcomes.
• Algebra: In the expansion of binomial expressions and solving series related problems.

These applications make the factorial of 9 a useful concept beyond theoretical math, extending into real-world scenarios and problem-solving.

• 1. Calculate the number of different ways to arrange 9 books on a shelf.
• 2. If 9 people are running in a race, how many different ways can the first three winners be chosen?

1. There are 9! or 362,880 different arrangements for the 9 books on a shelf.
2. For the race with 9 runners, the first three winners can be chosen in 9!/(9-3)! or 9! / 6! ways, which is 9 × 8 × 7, or 504 different ways.