how is ! used in math

The exclamation mark symbol (!) in mathematics is known as the factorial symbol. It is used in the context of counting or calculating the factorial of a non-negative integer.

The factorial of a non-negative integer n, denoted as n!, is the product of all positive integers from 1 to n. For example, 5! (read as "5 factorial") is calculated as:

5! = 5 x 4 x 3 x 2 x 1 = 120

The exclamation mark serves as a notation to indicate factorial, which simplifies the expression.

To calculate the factorial of a number n, you can use the following steps:

1. Start with the value of n.
2. Multiply that value by each positive integer less than it, all the way down to 1.
3. Continue multiplying until you reach 1.

In equation form, this can be represented as:
n! = n x (n-1) x (n-2) x ... x 2 x 1

For instance,
4! = 4 x 3 x 2 x 1 = 24
6! = 6 x 5 x 4 x 3 x 2 x 1 = 720

The factorial function is commonly used in combinatorial mathematics, permutations, and probability calculations.