Ten books numbered 1-10 are to be put onto three shelves numbered 1-3. How many

Question

Ten books numbered 1-10 are to be put onto three shelves numbered 1-3.
How many ways can this be done, if each shelf must have at least one book
and the ordering of the books on each shelf matters?
For example:
shelf 1: 2,8,4
shelf 2: 9,7,3
shelf 3: 1,10,5,6
The answer for this is 10! * (9 choose 2) and I am not sure if I understand this.

I was able to get the last (9 choose 2)part. Basically you remove 3 books and then you divide the rest of the 7 books in 3 shelves( _ _ _ |_ _ _ | _ and then this will be 9 choose 2). But I don't know how come there is a 10! in the answer. Thoughts

Explanation / Answer

you set up the dividers correctly. there are (9 choose 2) ways to order the books and the dividers. but remember that the order of the books matters. even after sorting the divider/book array, you still have to order the 10 books. hence, there is 10! ways of doing this. so the overall answer is 10! * (9 choose 2).

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