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

$$$5\cdot4\cdot3\cdot2\cdot1 = 120$$$

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".

$$$5!=5\cdot4\cdot3\cdot2\cdot1 = 120$$$

Other examples would be:

- Factorial of three: $$ 3! = 3\cdot2\cdot1 = 6$$
- Factorial of eight: $$8! = 8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1 = 40320$$
- Factorial of four: $$4! = 4\cdot3\cdot2\cdot1 = 24$$

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.

For example,: $$$8! = 8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1$$$ can also be written as: $$$8! = 8\cdot7 \cdot \ldots \cdot 2\cdot1$$$

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: $$$54! = 54\cdot53\cdot52 \cdot \ldots \cdot 3\cdot2\cdot1$$$

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: $$$ n!=n(n-1)(n-2)\cdots2\cdot1$$$

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: $$$n! = n \cdot (n-1)!$$$

For example, in the factorial of $$8$$

$$$8! = 8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1$$$

We can associate the factors as follows: $$$8 \cdot (7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1)$$$

the group in brackets is precisely $$7!$$. So that we can write down: $$$8! = 8 \cdot (7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1) = 8\cdot7!$$$

$$7! = 7\cdot6!$$

$$11! = 11\cdot10\cdot9!$$

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