How many ways can 5 different mathematics books 4 different science book and 3 different English books be arranged on a shelf?

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)

Page 2

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.

Page 3

* 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!

Page 4

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

Look for it... It will either be

(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:

Page 5

Question: "In how many ways can 2 different history books, 5 different math books, and 4 different novels be arranged on a shelf if the books of each type must be together?"

In this question, sequence of the books is not important, therefore:

For the 2 history books: 2 ways to arrange them (AB and BA), or $2!$
For the 5 math books: $5*4*3*2*1 = 5!$ ways to arrange them, or 120
For the 4 novels: $4*3*2*1 = $4!$ ways to arrange them, or 24

Think like this:

For the history books (assuming we only look at the history books): 2 options for the first slot, and 1 for the last
For the math books (again, only look at the math books): 5 options for the first slot, $5-1=4$ for the second slot, $5-2=3$ for the third and so on
The same for the novels

We also have three types of books, so, the order of first-to-appear is, by the same logic, 3!

Therefore, in the the end you have $2!*5!*4!*3!=34560$ ways to arrange those books