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Solutions
Given:-
Eight residents - Robert, John, David, Thomas, Smith, George, Harry and William
Floors - Four, Ground floor is numbered 1 and the Topmost floor is numbered 4.
Two Flats - A and B, Flat B is in the east of flat A.
Now,
1) There are two floors between David and Thomas, who lives above David and both of them live in different flats. So, the possibilities are -

2) George lives in flat A and lives above Harry, who lives above William. We don't know the flats of Harry and William so the possibilities are -
Case I -

Case II -
| Floors |
Case II (a) |
Case II (b) |
Case II (c) |
Case II (d) |
Case II (e) |
Case II (f) |
Case II (g) |
Case II (h) |
Case II (i) |
Case II (j) |
| Flat A |
Flat B |
Flat A |
Flat B |
Flat A |
Flat B |
Flat A |
Flat B |
Flat A |
Flat B |
Flat A |
Flat B |
Flat A |
Flat B |
Flat A |
Flat B |
Flat A |
Flat B |
Flat A |
Flat B |
| 4 |
George |
Thomas |
George |
Thomas |
George |
Thomas |
George |
Thomas |
|
Thomas |
|
Thomas |
George |
Thomas |
George |
Thomas |
George |
Thomas |
George |
Thomas |
| 3 |
Harry |
|
|
Harry |
|
Harry |
Harry |
|
George |
|
George |
|
|
|
|
|
Harry |
|
|
Harry |
| 2 |
William |
|
William |
|
|
William |
|
William |
Harry |
|
|
Harry |
Harry |
|
|
Harry |
|
|
|
|
| 1 |
David |
|
David |
|
David |
|
David |
|
David |
William |
David |
William |
David |
William |
David |
William |
David |
William |
David |
William |
3) Robert lives exactly in the west of John. It means Robert lives in flat A and John lives in flat Q and both lives on the same floor. This is not possible in any of the cases of Case I so Case I gets completely eliminated. In Case II, it is possible only in Case II (g), Case II (h), Case II (i), and Case II (j).
Case II -

4) Robert lives just above Smith and both of them live in the same flat. It means Smith also lives in flat A. Now, this is not possible in Cases (g), (i) and (j) of Case II so they get eliminated.
Thus, the final arrangement is -
Case II -

Among the given options, Smith lives on the second floor.
Hence, the correct answer is Smith.