The factorial of a number

Let's take a positive integer, for example $5$ , and let's do the following multiplication:

$5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120$

That is, the product of all the positive integers that are less than $5$ .

The result is called factorial of five and it is indicated by an exclamation mark next to the number five: $5!$ and it is read "factorial of five".

$5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120$

Example

Other examples would be:

Factorial of three: $3! = 3 \cdot 2 \cdot 1 = 6$
Factorial of eight: $8! = 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 40320$
Factorial of four: $4! = 4 \cdot 3 \cdot 2 \cdot 1 = 24$

All scientific calculators have a key that allows us to do this calculation. It is usually indicated with an $x!$ . Thus, if we want to calculate the factorial of a number, we must first write the number in the calculator and then press the key $x!$ .

When we are dealing with big numbers, the factorial expression is long and it is possible to cut it short by means of suspension points.

Example

For example,: $8! = 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$ can also be written as: $8! = 8 \cdot 7 \cdot \dots \cdot 2 \cdot 1$

Example

To write, for example, $54!$ it is enough to write a few numbers at the beginning and others at the end, separated by suspension points: $54! = 54 \cdot 53 \cdot 52 \cdot \dots \cdot 3 \cdot 2 \cdot 1$

We are now ready to give the general definition of a factorial of a number. The factorial of a positive integer $n$ is defined as: $n! = n (n - 1) (n - 2) \dots 2 \cdot 1$

Logically $1! = 1$ . It does not seem that logical that $0! = 1$ , but this is adopted as a convention. So, for the factorial calculation it is important to remember that $1! = 1$ and $0! = 1$ .

It is easy to observe, using a calculator, that the factorial of a number grows in an almost exponential way; in other words, it grows very quickly.

$10! = 3628800$

$15! = 1307674368000$

$20! = 2432902008176640000$

Therefore, it can be difficult to clear bothersome calculations when operating with factorials.

A property of factorials used to simplify fractions is: $n! = n \cdot (n - 1)!$

Example

For example, in the factorial of $8$

$8! = 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$

We can associate the factors as follows: $8 \cdot (7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1)$

the group in brackets is precisely $7!$ . So that we can write down: $8! = 8 \cdot (7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1) = 8 \cdot 7!$

Example

$7! = 7 \cdot 6!$

$11! = 11 \cdot 10 \cdot 9!$

$x! = x \cdot (x - 1) \cdot (x - 2) \cdot (x - 3)!$