|Spoiler Inside: Solution to Puzzle||SelectShow>|
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 4a.
The number of horror-romance-only books is then 2a + b. (a + b sf-romance books doubled minus b (the duplication)).
4a + a + b + (2a + b) = 300 (the number of romance books of all combinations). This simplifies to 7a + 2b = 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 + (2a + b) + (a + b) + 2b = 200 which simplifies to 4a + 4b = 200.
Solving the system 7a + 2b = 300 and 4a + 4b = 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.