# Ten books are to be arranged on a shelf. Permutation Question

Ten books are to be arranged on a shelf. Three of the books must be together (in any order)because they form a trilogy, and another two must be together in the correct order (volume 1,then volume 2). In how many ways can the books be arranged?

This is a question from my textbook and the answer says 51,840 which I do not know how they get.

My working is that, I consider the trilogy as one book and the pair of books also as one. Thus I thought that we only need to find how many ways we can arrange the 5 remaining books and multiply it with the number of ways we can get for the trilogy and the pair. Since the order does not matter for the trilogy, I used $3!$ and I said that it does not matter where the pair goes.

Thus my calculation was $5! \times 3! =720$ which is very off to the answer. Any help would be appreciated!

• $5$ books + $1$ pack of three + $1$ pair: $(5+1+1)! =7!$ arrangements
• the pack of three: $3!$ arrangements within the pack
All together: $$7! \cdot 3! = 30240$$