Here are the five books:

Let's use slots like we did with the license plates:

We'll fill each slot -- one at a time...  Then we can use the counting principle!

The first slot:

We have all 5 books to choose from to fill this slot.

Let's say we put book C there...

Now, we only have 4 books that can go here...
 

How many books are left for this slot?
 

See it?

Whoa, dude!  That's 5!

So, there are 120 ways to arrange five books on a bookshelf.
(Aren't you glad I didn't make you draw them out?)

Was the answer to our 3-book problem really 3! ?

Yep!

Will this always work?

TRY IT:

How many ways can eight books be arranged on a bookshelf?  (reason it out with slots)