How many ways can 3 books on mathematics and 5 books on English be placed so that books on the same subject always remain together?

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:

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

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)

Now, we're going to learn how to count and arrange.  (As if just learning to count wasn't exciting enough!)

How many ways can we arrange three books on a bookshelf?

     Here are the books:

Well, there's one arrangement.

Let's pound out the others:

That's all of them...  There are 6 ways to arrange three books on a bookshelf.

What about five books?

Dang!  I don't want to have to draw it all out!

Let's FIGURE it out instead.

* For this one, order does NOT matter!

We did this problem before:

     If we have 8 books, how many ways can we arrange 3 on a
     bookshelf?

We figured it out with slots:

But, using the formula gave us the same thing:

Here's a different question for you:

     If we have 8 books and we want to take 3 on vacation with us, how
     many ways can we do it?

What's the difference between these problems?

ORDER DOESN'T MATTER!

In the first problem, we were arranging the 3 books on a shelf...  and in the second problem, we're just tossing the 3 books in a suitcase.

So, if order doesn't matter, we'll just divide it out!

Arranging the 3 books is 3!

Grab a calculator!  I'm going to teach you about a new button.

Look for it...  It will either be

or

(It's probably above one of the other buttons.)

Find it?

It's called a factorial.

Here's an example:

(No, this isn't just an excited 5.)

Here's what it means:

Check it by multiplying it out the long way, then try the button.

Here are some others: