What is Factorial (n!)?

The factorial of a non-negative integer n, denoted as n!, 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.

The factorial function is used in many areas of mathematics, including combinatorics and probability. It has a number of important properties and applications, including the formula for the number of combinations and permutations, the Taylor series expansion of functions, and the solution of differential equations.

It is worth noting that the factorial function grows very rapidly. For example, 10! = 3628800, 20! = 2432902008176640000, and 50! is an enormous number with more than 60 decimal digits.

Factorial definition formula

The factorial of a non-negative integer n, denoted as n!, is defined as the product of all positive integers less than or equal to n. The formula for n! can be expressed as:

n! = n x (n-1) x (n-2) x … x 3 x 2 x 1

For example, 5! = 5 x 4 x 3 x 2 x 1 = 120.

It is worth noting that the factorial of 0, denoted as 0!, is defined as 1, since 0! = 0 x 1 = 1. This definition is important for many mathematical and statistical applications, as it provides a convenient starting point for various calculations.

Recursive factorial formula

A recursive formula for factorial n is a formula that defines n! in terms of the factorial of a smaller integer.

Here’s a recursive formula for the factorial of n:

n! = n x (n-1)! if n > 0

0! = 1

The formula says that the factorial of n is equal to n multiplied by the factorial of n-1, where the factorial of 0 is defined as 1. This recursive definition allows us to compute the factorial of any non-negative integer by repeatedly applying the formula, starting from the base case of 0! = 1.

Here’s an example of how this formula can be used to compute the factorial of 5:

5! = 5 x (5-1)! = 5 x 4! = 5 x (4 x (4-1)!) = 5 x 4 x 3! = 5 x 4 x (3 x (3-1)!) = 5 x 4 x 3 x (3-1) = 5 x 4 x 3 x 2 x 1! = 5 x 4 x 3 x 2 x 1 = 120

Recursive formulas like this can be implemented as a function in many programming languages, and they are a common way to solve problems that can be broken down into smaller subproblems. The recursive factorial formula is a simple example of a more general concept called recursion, which is a fundamental technique in computer science and mathematics.