In how many ways can eight books, including two of English, can be arranged in such a way that the English books are never together (i. e. next to each other)?
There are a total of $8! = 40320$ ways of arranging those eight books.
First, we consider the case when the two English books are next to each other. Then these two books may be regarded as constituting a single block, and this block can appear in seven different positions, thus making for seven mutually exclusive cases.
In each of these seven cases, there are two ways of arranging the two books on English within the block.
What next? How to proceed from here?