You want the number of combinations with the four blue books together. Choosing such a combination is equivalent to choosing:

- first, the 4 "blue locations" (there are 9 possibilities)

- then the order of the books in these "blue locations" (there are 4! possibilities)

- finally the order of the other books, in the other locations (there are 8! possibilities).

So you end up with combinations.

The "tip" here (and in such cases) was to split your problem into choices of locations and then choices of the ordering.

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What you have computed is , that is to say the number of ways to choose 4 elements in a set of 12. Indeed, there are 12! combinations of books, and by dividing by 4! you get the number of combinations of books when you group together (and count as 1) the combinations where the 4 blue books globally lie on the same positions (they come by 4!). Then, dividing by 4!8!, you get the number of combinations of 12 books, 4 blue and 8 other than blue, when you group together the combinations where the positions of the colors are the same. So you get the number of ways to choose 4 "blue" positions among 12 slots (no matter in which order).

So when you divide, it means you group together (or "identify") certain combinations, which was not the case here.

Laurent.