At no time was the 12th member mentioned, apart from in the first paragraph. The reason that this puzzle appears to work is because of the phrasing of the paragraph beginning He leads the party into the hotel
which leads you to believe that room 10 is eventually occupied by the 11th member of the party, leaving room 11 free to place the 12th member in, after taking him from room 1. In reality, members 1 & 2 are already in room 1, and it is one of those members that eventually ends up in room 11. The final member of the group, member 12, is NOT allocated a room and has to spend the night in the stables with the horses.