# number of ways to arrange books

I am solving number of arrangements for following question:

Eight books are placed on a shelf. Three of them form a 3-volume series, two form a 2-volume series, and 3 stand on their own. In how many ways can the eight books be arranged so that the books in the 3-volume series are placed together according to their correct order, and so are the books in the 2-volume series? Noted that there is only one correct order for each series.

I analyzed this problem in following way: 3 volume book as A, 2 volume book as B, and remaining books as C. So three can be arranged as 3!, and C in turn can be arranged as 3! which by product rule we can have 3! * 3! = 36. Why this analysis or approach to this problem is wrong. Correct answer is 120 (i.e. 5!). Kindly help. thanks

• The C books do not need to be next to each other. Sep 17, 2020 at 9:58

You can think of the three books who make a volume as a single big book (their relative positions can't change) and so for the other two volumes books, they can be considered as a single one too since their relative position can't change, therefore you have the equivalent of 5 books:

(3 books), (2 books), (1 book), (1 book), (1 book)

And the number of ways you can arrange them is $$5!$$ indeed.

Since the $$3$$-volume series are placed together according to their correct order, we have $$A_{1}A_{2}A_{3}$$ and for the $$2$$ volume series we also have $$B_{1}B_{2}.$$

So we can consider the $$3$$ volume books to be a single book denoted by $$A$$ and the $$2$$ volume books to be a single book denoted by $$B$$. Thus we need to arrange $$A,B$$ and the remaining $$3$$ books. This is the same as the number of ways to rearrange $$5$$ books.

Since the number of ways in which $$5$$ books can be arranged is $$5!=120,$$ we have $$120$$ ways.

• thanks for explanation. Can you point out what is wrong in my analysis Sep 17, 2020 at 11:24
• As mentioned in the comments, you calculated the $C$ books to be next to each other, but they do not need to be next to each other. Sep 17, 2020 at 11:26