2025 TJC P2 Q6

A treasure hunt game is to be played in a \(3\)-storey building. There are \(12\) specific locations where \(5\) treasures are to be hidden. Among the \(5\) treasures, \(4\) of them are identical. The locations are distributed across the storeys as follows:

• \(3\) locations on the \(1\)st storey,
• \(3\) locations on the \(2\)nd storey, and
• \(6\) locations on the \(3\)rd storey.

Each location can hold at most one treasure.

1. Find the number of ways to distribute the \(5\) treasures among the \(12\) locations for each of the following separate cases:
   1. The \(5\) treasures can be hidden in any of the \(12\) locations.[1]
   2. Exactly \(2\) treasures must be hidden on the \(3\)rd storey.[2]
   3. There must be at least one treasure on each of the \(1\)st and \(2\)nd storeys, and at least \(2\) treasures on the \(3\)rd storey.[3]

Five teams of \(2\) players each take part in the game. The \(10\) players and \(2\) game masters are to be seated around a round table for a briefing.

2. Find the number of different seating arrangements such that players from the same team must be seated next to each other but the two game masters must not be seated next to each other.[3]