There are 120 different arrangements. The first way is to put all the books on the shelf. Then, every time you want to rearrange them, you can pick up one book and move it to another spot.
The second way is to put all the books on the shelf except for one book, which you will place at the top of the shelf. Then, every time you want to rearrange them, you can pick up a book from either end of the shelf and move it towards the other end of the shelf.
The third way is to put four books in one row and leave two empty slots in between each pair of books on either side (see illustration). You can then pick up any book from this row and move it around into any of those empty slots without disturbing any other books in that row.
Each time you rearrange your books using these three methods (or some combination thereof), there are 120 possible arrangements left over from before — so there are 120 possible ways in which six books can be arranged on a shelf with no repetition of any particular arrangement!
Last modified: August 14, 2022