By opening rooms at a higher level, you can get more tickets for items on the search list. In addition, the higher the skill level in the room, the more items will be on your search list — and, accordingly, the more tickets you can get.
To get even more tickets, you may use an amulet which increases your chances to find an item.
To get even more tickets, you may use an amulet which increases your chances to find an item.