# How Factorials Work

Theory

### Factorial

Factorial is an operation that multiplies all the numbers from $n$ down to 1 with each other. The symbol for factorial is “!”.

 $n!=1\cdot 2\cdot 3\cdots n$

Example 1

Four factorial is 24, because 1 times 2 times 3 times 4 is 24. It looks like this with symbols:

 $4!=4\cdot 3\cdot 2\cdot 1=24$

The reason why the factorial is so important, is that it tells you how many different ways you can order a set of $n$ elements.

Example 2

You have the three letters “A”, “B” and “C”, and you wonder how many different words you can make with these if you can only use each letter exactly once. Since you make a word by ordering the letters, there are $3!=3\cdot 2\cdot 1=6$ different words.

Another way of thinking is as follows: The first letter can be chosen in three different ways, because there are three letters. After you have chosen the first letter, there are two left, so the second letter can be chosen in two ways. When you have chosen the second letter, there is only one letter left, so you only have one choice for the last letter. To get the number of possible “end results”, you multiply the number of choices in each step, and you get $3\cdot 2\cdot 1=6$.

If you’re in doubt, you can try doing it with pen and paper and see that it’s actually correct that there are precisely six words:

1.
ABC
2.
ACB
3.
BAC
4.
BCA
5.
CAB
6.
CBA

Example 3

A school class is choosing the seating arrangements for the classroom. There are 20 students in the class, and there are 20 desks. In how many ways can the students be seated at the desks?

In this case it might be smart to think of it like this:

When the occupant of the first desk is being decided, how many different students can be put there? There are 20 students in the class, so there are 20 possible students to choose from.

When the occupant of the second desk is being decided, how many different students can be seated at this one?

There are 19 students left because 1 has already been seated. You can ask this question until you reach the last student. That means the calculation becomes

 $20!=20\cdot 19\cdot 18\cdots 2\cdot 1\approx 2.4\cdot 1{0}^{18}.$

Example 4

A handball team is going on tour and is deciding who gets to sleep in which bed at the hotel. There are 14 players on the team. In how many ways can the coaches distribute the beds between the players?

When choosing which player gets the first bed, the coaches have 14 players to choose from. For the second bed, they have 13 players to choose from, and for the third bed they have 12 players to choose from. They continue like this until they’re left with one bed and one player. The calculation becomes:

 $14!=14\cdot 13\cdot 12\cdots 2\cdot 1=87\phantom{\rule{0.17em}{0ex}}178\phantom{\rule{0.17em}{0ex}}291\phantom{\rule{0.17em}{0ex}}200.$