In how many ways can you place 9 different books on a shelf if there is space enough for 5 books

You can divide your task in steps, count the ways that you can conduct each step and then use the multiplication principle to count the total number of ways that you can conduct your task.

a) You can do it in two steps.

1.Step. Order your books. You can do it in $25!$ ways.

2.Step. Decide on how many books will be placed on each shelf. That is equivalent to writing on piece of paper 10 nonnegative (zeros are here allowed in c they are not!) integers $x_1,x_2,\dots,x_{10}$ so that they add up to 25. For $i=1,2,\ldots,10$, $x_i$ represents the number of books that will be placed on the $i$-th shelf. The total number of ways that you can conduct this step is equal to the number of solutions to the equation $$x_1+x_2+\ldots+x_{10}=25,$$ with $0\leq x_i, i=1,2,\ldots,10$. This is a typical stars and bars problem with $n=10$ and $k=25$. Then (according to the Wikipedia page, Stars and Bars, Theorem two) you can do that in $$\dbinom{n+k-1}{n-1}=\dbinom{10+25-1}{10-1}=\dbinom{34}{9}$$ So in sum the task in (a) can be conducted in $25!\times\dbinom{34}{9}$ ways.

c). Based on a) the answer is straightforward by allowing in Step 2. only positive values for the $x_i, i=1,2,\ldots,10$. The total number of ways that you can conduct this step is equal to the number of solutions to the equation $$x_1+x_2+\ldots+x_{10}=25,$$ with $1\leq x_i, i=1,2,\ldots,10$. Then (according to the Wikipedia page, Stars and Bars, Theorem one) you can do it in $\dbinom{k-1}{n-1}=\dbinom{25-1}{10-1}$ ways.

So in sum the task in (c) can be conducted in $25!\times\dbinom{24}{9}$ ways.

b) 25C25 is equal to 1 so, your approach here is not correct. Here you can think it as follows. Write on each book a number from 1 to 10 (this number indicates the shelf that it will be placed). So you can do it $10\times10\times\ldots\times10=10^{25}$ ways. (With steps as in the previous questions: 1.Step Choose a shelf for book Nr1. You can do it in 10 ways. 2.Step Choose a shelf for book Nr2. You can do it in 10 ways and so on until 25.Step Choose a shelf for book Nr25. You can do it in 10 ways. So by multiplication rule you can do it in $10^{25}$ ways.)

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)

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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.

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* 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!

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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:

                                            

                                            

                                            

                                             

                                                

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