Question

A bookshelf contains 3 Russian novels, 4 German novels and 5 Spanish novels. All the Spanish novels must be together and no constraint on the order of the others. How many ways can you arrange these books?

Answer 1

`5!xx8xx7! = 120xx8xx5040=4,838,400` ways

Explanation:

Spanish Novels - internal ordering

The novels must be kept together and so we'll have them be a block of books on the shelf.

How many ways can we order the Spanish novels? `5!`

Spanish Novels - placement

We now have a block of 5 books to place on the shelf. There are 12 books in total, and so 12 spots for books. The block of Spanish novels can be in spots 1 - 5, 2 - 6, 3 - 7, 4 - 8, 5 - 9, 6 - 10, 7 - 11, 8 - 12, or 8 spots in total.

The other novels

We can place the 7 remaining novels in the remaining 7 spots in `7!` ways.

All together

And so all told, we can arrange the books:

`5!xx8xx7! = 120xx8xx5040=4,838,400` ways

(If this seems like a lot, keep in mind that without the constraint on the Spanish novels, we could have arranged the books in `12! = 479,001,600` ways