M, N, O, P, Q, R, and S live on seven different floors of a building numbered from 1 (lowest) to 7 (highest). Each person has a distinct income from the set \{3500, 15000, 7500, 9000, 11000, 13500, 5000\}, but not necessarily in the same order.

Given clues:

- M lives on an odd-numbered floor but not on floor 3.
- The person earning 11000 lives immediately above M.
- Exactly two people live between M and the person earning 7500.
- The person earning 15000 lives on an odd-numbered floor above P.
- Exactly three people live between O and the person earning 15000.
- The person earning 7500 lives immediately above O.
- R earns 4000 more than Q.
- The person earning 3500 lives immediately above the person earning 5000.
- Exactly one person lives between N and Q, and N lives above Q.
- Neither O nor M earns 9000.
- Q does not earn 7500.

**Question:** Which of the following income-person combinations is correct based on the above arrangement?

Question

M, N, O, P, Q, R, and S live on seven different floors of a building numbered from 1 (lowest) to 7 (highest). Each person has a distinct income from the set \{3500, 15000, 7500, 9000, 11000, 13500, 5000\}, but not necessarily in the same order.

Given clues:

- M lives on an odd-numbered floor but not on floor 3.
- The person earning 11000 lives immediately above M.
- Exactly two people live between M and the person earning 7500.
- The person earning 15000 lives on an odd-numbered floor above P.
- Exactly three people live between O and the person earning 15000.
- The person earning 7500 lives immediately above O.
- R earns 4000 more than Q.
- The person earning 3500 lives immediately above the person earning 5000.
- Exactly one person lives between N and Q, and N lives above Q.
- Neither O nor M earns 9000.
- Q does not earn 7500.

**Question:** Which of the following income-person combinations is correct based on the above arrangement?

Accepted Answer

**Solution:**

1. M lives on an odd floor but not floor 3, so M can be on floor 1, 5, or 7.

2. The person earning 11000 lives immediately above M, so M cannot be on floor 7 (no floor above).

3. Only two people live between M and the person earning 7500.

4. The person earning 15000 lives on an odd floor above P.

5. Exactly three people live between O and the person earning 15000.

6. The person earning 7500 lives immediately above O.

7. R earns 4000 more than Q.

8. The person earning 3500 lives immediately above the person earning 5000.

9. Exactly one person lives between N and Q, with N above Q.

10. Neither O nor M earns 9000.

11. Q does not earn 7500.

**Step-by-step deduction:**

- Since M is on an odd floor (not 3), possible floors: 1, 5, 7.

- The 11000 income is immediately above M, so M cannot be on 7. If M is on 5, 11000 is on 6; if M is on 1, 11000 is on 2.

- There are exactly two people between M and the one earning 7500. So if M is on 1, 7500 is on 4; if M is on 5, 7500 is on 2.

- The person earning 7500 lives immediately above O, so O is one floor below 7500.

- If 7500 is on 4, O is on 3; if 7500 is on 2, O is on 1.

- The person earning 15000 lives on an odd floor above P.

- Exactly three people live between O and the one earning 15000.

- If O is on 3, then 15000 is on 7 (3 + 4 = 7).

- If O is on 1, then 15000 is on 5 (1 + 4 = 5).

- Since 15000 must be on an odd floor above P, and 15000 is on 7 or 5, P must be below that floor.

- R earns 4000 more than Q.

- The person earning 3500 lives immediately above the one earning 5000.

- Exactly one person lives between N and Q, with N above Q.

- Neither O nor M earns 9000.

- Q does not earn 7500.

**Assigning floors and incomes:**

- Let’s try M on floor 1:
  - 11000 on floor 2.
  - 7500 on floor 4.
  - O on floor 3 (immediately below 7500).
  - 15000 on floor 7 (3 + 4 = 7).
  - P must be below 7.

- The person earning 3500 lives immediately above the one earning 5000.

- N and Q have one person between them, with N above Q.

- R earns 4000 more than Q.

- Neither O nor M earns 9000.

- Q does not earn 7500.

**Final arrangement:**

| Floor | Person | Income |
|-------|--------|--------|
| 7     | R      | 15000  |
| 6     | S      | 9000   |
| 5     | N      | 13500  |
| 4     | P      | 7500   |
| 3     | O      | 5000   |
| 2     | M      | 11000  |
| 1     | Q      | 3500   |

- R earns 15000.
- N earns 13500.
- O earns 5000.
- M earns 11000.
- P earns 7500.
- Q earns 3500.
- S earns 9000.

- R earns 4000 more than Q (15000 - 3500 = 11500, so this contradicts the clue; so adjust N and S incomes accordingly).

- Adjusting R and Q:
  - If Q earns 11000, R earns 15000 (difference 4000).
  - But M earns 11000, so Q cannot earn 11000.

- Try Q = 7500 (not allowed), so Q = 5000, R = 9000.

- O earns 13500 (since O is not 9000 or 11000).

- M earns 11000.

- P earns 3500.

- S earns 15000.

- N earns 7500.

- Final correct combination is 13500 - O.

**Answer:** (a) $13500 - O$

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