The factorial of a number

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: $$8! = 8 \cdot (7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1) = 8\cdot7!$$

$$7! = 7\cdot6!$$

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

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