Permutations & Combinations: Treasure; round table seating — TJC 2025 H2 Math Prelim Paper 2
What this question tests
Question
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 1st storey,
- 3 locations on the 2nd storey, and
- 6 locations on the 3rd storey.
- Find the number of ways to distribute the 5 treasures among the 12 locations for each of the following separate cases:
- The 5 treasures can be hidden in any of the 12 locations.
- Exactly 2 treasures must be hidden on the 3rd storey.
- There must be at least one treasure on each of the 1st and 2nd storeys, and at least 2 treasures on the 3rd storey.
- 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. (Five teams of 2 players each, plus 2 game masters, are seated around a round table.)
Show full worked solution▾
(a)(i) Choose 5 locations from 12 for the treasures (4 identical, 1 distinct): \[ \binom{12}{5}\binom{5}{1} = 792 \times 5 = 3960 \]
(a)(ii) Exactly 2 on the 3rd storey: choose 2 from 6 locations on 3rd storey, then place remaining 3 treasures in the other 9 locations (\(3+3=6\) on 1st and 2nd), choosing the 1 distinct treasure's position: \[ \binom{6}{2}\binom{6}{3}\binom{5}{1} = 15\times 20\times 5 = 1500 \]
(a)(iii)
Number of ways to arrange the 10 participants \(= (5-1)!(2!)^5 = 768\)
Number of ways to slot in the 2 game masters \(= \dbinom{5}{2}2! = 20\)
Required number \(= 768 \times 20 = 15360\)
(b) Treat each of the 5 teams of 2 as a unit. Arrange 5 team-units around a round table: \((5-1)! = 24\) ways. Within each team, 2 arrangements: \((2!)^5 = 32\). So \(24\times 32 = 768\) arrangements for the 10 players.
Now slot the 2 game masters into the 5 gaps between team-units (no two adjacent game masters): choose 2 of 5 gaps and arrange: \(\binom{5}{2}\times 2! = 10\times 2 = 20\).
Required number \(= 768\times 20 = 15360\).