Factorial Properties

The factorial of a non-negative integer n is denoted by n! and is the product of all positive integers less than or equal to n. For example, 5! = 5 x 4 x 3 x 2 x 1 = 120.

Factorial Properties:

1. Product Property: The factorial of a number is the product of that number and the factorial of the number minus 1. This can be expressed as n! = n x (n-1)!
2. Identity Property: The factorial of 0 is 1, represented as 0! = 1. This serves as the base case for many factorial calculations.
3. Recursive Property: Factorials can be calculated recursively using the product property. For example, 5! = 5 x 4!, 4! = 4 x 3!, and so on.
4. Factorial of a Negative Number: The factorial of a negative number is undefined and is not considered in the factorial function.
5. Factorial of a Fraction or Decimal: Factorials are defined for non-negative integers only, so factorial of a fraction or decimal is not considered in the factorial function.

Factorial Notation:

Factorials are often denoted using the exclamation mark, such as 5! for the factorial of 5. This notation is commonly used in mathematics and computer programming.

Applications of Factorials:

Factorials have numerous applications in combinatorics, probability, and permutation calculations. They are also used in the computation of binomial coefficients and in the Taylor series expansion of exponential functions.

Understanding the properties of factorials is important for solving problems in various mathematical and computational contexts.

Feel free to use this study guide to review and practice problems related to factorial properties!