The Assembly Queue
Five students - Arnel, Bea, Carlo, Divina, and Elmer - must arrange themselves in a single-file queue for a school assembly. The student council has set four rules. (1) Bea refuses to stand first or last. (2) Carlo must stand immediately ahead of Divina, with nobody between them. (3) Arnel must stand somewhere ahead of Bea. (4) Elmer must stand last. A teacher looks at the rules and says, "There is more than one way to arrange you, so you still have a real choice to make." The students want to know exactly which arrangements the rules permit before anyone moves. All five names are used once each.
1. How many different queues satisfy all four rules? List every valid queue from front to back.
Read the worked answer
Step 1 - Pin what is fixed. Rule 4 puts Elmer at position 5. Only positions 1–4 remain.
Step 2 - Handle the linked pair. Rule 2 makes Carlo+Divina a single block, in that order (CD). Inside positions 1–4 the block can sit at (1,2), (2,3), or (3,4).
Step 3 - Fill the gaps. The two leftover slots go to Arnel and Bea. Rule 3 (Arnel ahead of Bea) decides which way round they go, and rule 1 must then be checked.
| CD block | Free slots | A/B order | Queue | Bea legal? |
|---|---|---|---|---|
| 1–2 | 3, 4 | A=3, B=4 | C D A B E | Yes (4th) |
| 2–3 | 1, 4 | A=1, B=4 | A C D B E | Yes (4th) |
| 3–4 | 1, 2 | A=1, B=2 | A B C D E | Yes (2nd) |
Answer: 3 valid queues - C D A B E; A C D B E; A B C D E.
2. The council later adds a fifth rule: Carlo must stand first. Using the three queues you already found, what is the full queue now, and who stands fourth?
Read the worked answer
Test each surviving queue against the new rule rather than re-deriving from scratch - that is faster and less error-prone.
- C D A B E → Carlo is 1st. ✔
- A C D B E → Carlo is 2nd. ✘
- A B C D E → Carlo is 3rd. ✘
Only one queue survives: C D A B E. Reading positions: 1 Carlo, 2 Divina, 3 Arnel, 4 Bea, 5 Elmer.
Answer: the queue is Carlo, Divina, Arnel, Bea, Elmer; Bea stands fourth.
Note this is consistent with the original rules - Divina is still immediately behind Carlo, Arnel is still ahead of Bea, Bea is neither first nor last, and Elmer is still last.
3. Ignoring the fifth rule, is there any student besides Elmer whose position is fixed by rules 1–4? Defend your answer with the three queues.
Read the worked answer
A position is fixed only if it is the same in every valid queue, so compare the three.
| Student | Position in C D A B E | in A C D B E | in A B C D E | Fixed? |
|---|---|---|---|---|
| Arnel | 3 | 1 | 1 | No |
| Bea | 4 | 4 | 2 | No |
| Carlo | 1 | 2 | 3 | No |
| Divina | 2 | 3 | 4 | No |
| Elmer | 5 | 5 | 5 | Yes |
Every student except Elmer occupies at least two different positions across the valid queues, so the rules constrain relationships (Carlo before Divina, Arnel before Bea) but do not pin anyone down.
Answer: no. Only Elmer's position is fixed - he is last in all three queues. A tempting wrong move is to assume Arnel 'must' be second or Bea 'must' be fourth because one queue happens to show that; a single valid queue is a witness, not a proof.