Factorial of a Number

Factorial | Comprehensive Guide

The factorial of a non-negative integer is a mathematical operation where you multiply the number by all of the positive integers smaller than it. Factorials are widely used in areas like permutations, combinations, and probability. It is represented by an exclamation mark (!).

What is Factorial?

The factorial of a number, denoted as n!, is the result of multiplying all positive whole numbers less than or equal to that number.

For example:

• 5! (read as "five factorial") is calculated as:
  5 × 4 × 3 × 2 × 1 = 120
• 3! is:
  3 × 2 × 1 = 6

Factorials are often used to count how many ways something can be arranged, among other applications.

Key Properties of Factorials

Factorial of 0:

• By definition, 0! equals 1. This is a special case in mathematics and is used to ensure consistency in various calculations.

Rapid Growth:

• Factorials grow extremely fast. For example:
  • 10! equals 3,628,800
  • 20! is a massive 2,432,902,008,176,640,000
    As the numbers increase, the factorial results quickly become very large.

Recursive Nature:

• Factorials can be defined in a step-by-step way, where each number's factorial depends on the factorial of the number before it. For example, 5! can be calculated as 5 × 4!, 4! as 4 × 3!, and so on.

Applications of Factorials

Permutations and Combinations:

• Factorials are commonly used to calculate permutations (the number of ways to arrange things) and combinations (the number of ways to choose things from a set).

Probability:

• Factorials are often found in probability theory, especially in cases where the arrangement of items matters.

Mathematics and Computer Science:

• Factorials are used in solving problems related to recursive algorithms, combinatorics, and complex mathematical expressions.

Factorials play a fundamental role in mathematics, with applications spanning various fields, including statistics, computer science, and operations research. Understanding how to calculate and apply factorials is essential for tackling a wide range of mathematical problems.

For more details and further examples, check out the full article on GeeksforGeeks: https://www.geeksforgeeks.org/factorial/.