Factorials

Background

There are many different ways multiplication can be used. One way is to look at the numbers of ways things can be arranged...

Question

In how many ways can 3 distinguishable books be arranged in a row on a shelf? How about 4 books?

Answer

3! = 6 ways

4! = 24 ways 

Analysis

Let's start looking at this topic by looking at the three books. Let's say they are a book each on Accounting, Biology, and Chemistry (and so we have books A, B, and C). How can we put them on the shelf?

Let's take book A and put it on the shelf first. I can now choose to put either book B or book C next to it. Let's do book B, which means that book C has to be the last book I put up. That gives the arrangement on the shelf of:

ABC

Now, I could have chosen to put book C next to A and get this arrangement:

ACB

That's 2 arrangements starting with A. I could have started with book B and gotten these two options:

BAC
BCA

and the same with starting with book C:

CAB
CBA

That's 6 choices in total for how I can arrange the books.

Let's take a closer look to see what's going on.

For each starting book - like A - I had two ways to arrange the other two books. And I can start with three different books. And so I can say that the number of ways to arrange the books is:

3 x 2 = 6

I'm going to go one step further and include a x 1 in there - it won't change the outcome but it will help in more complicated problems:

3 x 2 x 1 = 6

When I have a problem where I am putting distinguishable objects (objects I can tell apart) and I am putting them into a distinguishable order (such as putting them into a row), I can use this type of math to find the number of ways I can order them.

This is called a "factorial" (and yes, this is where the name for this resource came from!)

In fact, this type of math is so common that we have notation to show that we'll using it. We use an exclamation mark to show that we're going to multiply natural numbers up to the number indicated. In this case, we can say that:

3 x 2 x 1 = 3! = 6

In general, we use the notation of n! for factorial calculations.

So now let's look at what happens if we add a 4th book to the mix, book D. Let's put D on the shelf first - now see that there are 6 ways to arrange books A, B, and C (what we just found above). Now let's put A in front - there are 6 ways to arrange books B, C, and D. And we can do the same with B and C as the first book. So in total, we have:

4 x 6 = 4 x 3! = 4 x 3 x 2 x 1 = 4! = 24

Vocabulary used:

For more information check out these links (comment to add your favourite link):

Where might you have come from?

Fact-orials Index

Operations:

• Multiplication

Where might we go?

Combinatorics:

• Factorials in a Circle
• Factorials with Identical Items
• Permutations