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Assuming that each of the nine books is unique, that there are only nine positions open on the given shelf, and that each book can fit in each position, the answer is 9! (nine factorial) which is equal to 362,880.

If the nine books are not all unique (i.e. there are two of the same book), the number should decrease taking into effect that reversing the positions of the identical books makes no overall change.

If there is one additional position, the number should increase to 10! or 3,628,800. If there is more than one additional position, the number will still increase, but not to 11! or 12! since both open spaces are reversible just like the identical books.

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