Correct Option (D)
The problem requires ensuring that any two consecutive floors have different colours. Let the four distinct colours be C1, C2, C3, and C4. For simplicity, let C1 be Red (R).
Consider Statement 3 along with Statement 2:
- Statement 3: The first (I) and the fifth (V) floors are painted Red (R).
- Statement 2: The second (II) and the fourth (IV) floors are painted in different colours.
From Statement 3, we have F_I = R and F_V = R.
To satisfy the condition that consecutive floors have different colours:
- F_II must not be R (since F_I = R).
- F_IV must not be R (since F_V = R).
Thus, F_II and F_IV must be chosen from the remaining three colours (C2, C3, C4). Let these be Blue (B), Green (G), and Yellow (Y).
Statement 2 specifies that F_II and F_IV are painted in different colours. Therefore, F_II and F_IV must be two distinct colours from {B, G, Y}. For example, we can assign F_II = B and F_IV = G.
The current arrangement is: F_I(R), F_II(B), F_III(?), F_IV(G), F_V(R).
Now, we need to determine F_III such that F_II ≠ F_III and F_III ≠ F_IV. This means F_III must not be B and F_III must not be G.
The available colours for F_III are R and Y. Since both R and Y are distinct from B and G, we can choose either of them for F_III. For instance, if F_III = R, the complete sequence is R, B, R, G, R. This sequence satisfies the condition that all consecutive floors have different colours (R≠B, B≠R, R≠G, G≠R).
Since a valid colouring can always be constructed under these combined conditions, Statement 3 along with Statement 2 is sufficient.
Incorrect Options:
Statement 1: "The middle three floors (II, III, IV) are painted in different colours."
This statement ensures that F_II ≠ F_III, F_III ≠ F_IV, and F_II ≠ F_IV. However, it provides no information about the colours of Floor I or Floor V relative to their adjacent floors. For example, if F_I is painted the same colour as F_II, or F_V is painted the same colour as F_IV (e.g., I=Blue, II=Blue, III=Green, IV=Yellow, V=Yellow), the condition of consecutive floors being different is violated. Therefore, Statement 1 alone is not sufficient.
Statement 2: "The second (II) and the fourth (IV) floors are painted in different colours."
This statement only ensures F_II ≠ F_IV. It does not provide any constraints for other consecutive floor pairs, such as F_I and F_II, F_II and F_III, F_III and F_IV, or F_IV and F_V. For example, if F_II and F_III are painted the same colour (e.g., I=Red, II=Blue, III=Blue, IV=Green, V=Yellow), the condition is violated. Therefore, Statement 2 alone is not sufficient.
Statement 3: "The first (I) and the fifth (V) floors are painted red."
This statement implies F_I = Red and F_V = Red. For consecutive floors to be different, F_II cannot be Red, and F_IV cannot be Red. However, this statement does not prevent other consecutive floors from being the same colour. For example, consider the sequence: F_I=Red, F_II=Blue, F_III=Blue, F_IV=Green, F_V=Red. Here, F_II and F_III are painted the same colour, violating the condition. Therefore, Statement 3 alone is not sufficient.
Statement 1 along with Statement 2:
Statement 1 already implies that F_II, F_III, and F_IV are all distinct colours, which inherently means F_II ≠ F_IV. Thus, Statement 2 adds no new information to Statement 1. As established, Statement 1 alone is not sufficient because it does not constrain the colours of Floor I and Floor V in relation to their neighbours. Hence, this combination is also not sufficient.