How many ways can you arrange books of the same kind of there are 4 math books 3 science books and 3 English books on a shelf?

As we arrange books on a bookshelf, there are a few different types of counting problems. Let�s take a look at a few of them.

Example 1:

�If there are three math books, five science books, and four English books, how many ways can they be arranged on a shelf if all the books are different books?�

There are a total of twelve books to place on the shelf. Picture twelve places on our shelf, which will be represented using blanks below.

There are twelve possible books for our first space on the shelf.

Once chosen, there are eleven books to use for the second space on the shelf.

Since two books have been shelved, there are ten books that can now be placed on the shelf.

We will continue this pattern until all books are used, which gives us these numbers.

The Fundamental Counting Principle requires us to multiply these numbers. Multiplying them is the same as�

ideo: Counting: Books on a Shelf Problems

uiz: Counting: Bookshelf Problems

Example 2:

�If there are three math books, five science books, and four English books, how many ways can they be arranged on a shelf if all the books are different books and we arrange them by subject?�

First, we have to realize that there are math books to place on the shelf. We can use blank spots to represent the books, like so.

If we place one book on the shelf at a time, we can see there are 3 possible math books we could place on the shelf, first.

Once a book is on the shelf, there are 2 possible books to place next on the shelf.

This leaves only 1 book to place last on the shelf.

We multiply them, due to the Fundamental Counting Principle and get 3!.

Likewise, when we place science books on the shelf, there are 5! Ways to do so. There are 4! ways to place the English books on the shelf.

However, we have to consider how many ways we can now place these books, by subject.

Since there are three subjects (math, English, and science), there are 3! ways to arrange them on the shelf. Here are those arrangements:

There are 3! ways or six ways to arrange the subjects.

Putting this all together, we get this is our answer:

...which is 103,680 ways to arrange all the books by subject.

ideo: Counting: Books on a Shelf Problems

uiz: Counting: Bookshelf Problems

In how many ways can $6$ English books, $4$ science books, $7$ magazines, and $3$ mathematics books be arranged on a shelf if English books are indistinguishable and science books are indistinguishable?

We have a total of $6 + 4 + 7 + 3 = 20$ books. Choose six of the $20$ positions for the English books and four of the remaining $14$ positions for the science books. The remaining ten positions can be filled with books and magazines in $10!$ ways.

$$\binom{20}{6}\binom{14}{4}10! = \frac{20!}{6!14!} \cdot \binom{14!}{4!10!} \cdot 10! = \frac{20!}{6!4!}$$

In your attempt, you did not take into account the total number of positions on the shelf.

In how many ways can $6$ English books, $4$ science books, $7$ magazines, and $3$ mathematics books be arranged on a shelf if English books should be together?

If all the books are intended to be distinct (switching the order of the questions would have made this clearer), treat the English books as a single object, so we have $1 + 4 + 7 + 3 = 15$ objects to arrange. Then multiply by the number of ways of arranging the six English books within the block of English books.

If we are still supposed to treat the English books as being indistinguishable and the science books as being indistinguishable, choose six of the $15$ positions for the science books, one of the remaining $8$ positions for the block of English books, then arrange the magazines and mathematics books in the remaining positions.

I believe the first of these two interpretations is intended, but I would have reversed the order of the questions to make that clear.