The other tutor answered the problem as though the keys were arranged in a row; that is certainly one interpretation.
9! = 362880
Another possibility is that the key chain is circular; then the problem is like arranging 9 people around a table. The number of arrangements in that case is
9!/9 = 8! = 40320
Additionally, most keys have a flat side and a jagged side, so each key could be on the chain in either of two ways. That would make the number of arrangements
(8!)(2^9) = 20643840
Or if the key chain is not circular and each key can be on the chain in either of two ways, the total number of arrangements is
(9!)(2^9) = 185794560
If this is actually a problem that you are supposed to solve, ask the person who asked you the question what the groundrules are....
