Let

*a* be the number of sf-romance-only books. Let

*b* be the number of sf-horror-romance books.

The number of romance-only books is then 4*a*.

The number of horror-romance-only books is then 2*a* + *b*. (*a* + *b* sf-romance books doubled minus *b* (the duplication)).

4*a* + *a* + *b* + (2*a* + *b*) = 300 (the number of romance books of all combinations). This simplifies to 7*a* + 2*b* = 300.

The number of sf-horror-only is *a* + *b* (the same as sf-romance-and-not-horror).

There are 1000 books, but 1200 by the three categories. The overlapping eliminates 200. That means that each of the two-category combinations counts twice, and the three-category combination counts three times. The amounts over one account for 200. Therefore *a* + (2*a* + *b*) + (*a* + *b*) + 2*b* = 200 which simplifies to 4*a* + 4*b* = 200.

Solving the system 7*a* + 2*b* = 300 and 4*a* + 4*b* = 200 gives *a* = 40 and *b* = 10. Applying these solves five of the combinations. Subtracting the more-than-one categories gives 400 sf-only and 250 horror-only.

The solution is 400 sf-only, 250 horror-only, 160 romance-only, 40 sf-romance-only, 90 horror-romance-only, 50 sf-horror-only, and 10 sf-horror-romance.