Which among the following statement(s) is/are true?

I. T lives above R II. Q lives on an odd numbered floor

III. R lives on the adjacent floor of Q

Study the following information carefully and answer the questions given below:

Six persons –P, Q, R, S, T and U are living on different floors of the six-storey building such that the lowermost floor is numbered as one and the floor immediately above it is numbered as two and so on. Each of them likes different colours- Red, Blue, Grey, Black, Pink and White but not necessarily in the same order. Only two floors are between the one who likes grey colour and U who lives on an odd numbered floor. P lives immediately above U. The number of floors below P is one more than the number of floors above the one who likes white. Only one floor is between Q and the one who likes pink colour. The one who likes Pink colour and the one who likes blue colour live on the adjacent floor. Neither U nor P likes blue colour. The number of floors between the one who likes black and S is two less than the number of floors between S and the one who like red. T likes neither grey nor red.

To find the correct statements, we can determine the unique floor and colour arrangement for the six persons (P, Q, R, S, T, and U) on the six floors (numbered 1 to 6 from bottom to top) based on the given clues:

1. U lives on an odd-numbered floor, and P lives immediately above U.
This gives three possible cases:
- Case A: U is on Floor 5, P is on Floor 6.
- Case B: U is on Floor 3, P is on Floor 4.
- Case C: U is on Floor 1, P is on Floor 2.

2. "Only two floors are between the one who likes grey and U."
- In Case A (U on Floor 5), the grey-liking person must be on Floor 2.
- In Case B (U on Floor 3), the grey-liking person must be on Floor 6.
- In Case C (U on Floor 1), the grey-liking person must be on Floor 4.

3. "The number of floors below P is one more than the number of floors above the one who likes white."
- In Case A: P is on Floor 6. Floors below P = 5. So, floors above White = 4. This means the white-liking person must be on Floor 2. But Floor 2 is already occupied by the grey-liking person. Thus, Case A is eliminated.
- In Case B: P is on Floor 4. Floors below P = 3. So, floors above White = 2. This means the white-liking person must be on Floor 4 (which is P). This is valid. Here, Grey is on Floor 6, and White is on Floor 4 (P).
- In Case C: P is on Floor 2. Floors below P = 1. So, floors above White = 0. This means the white-liking person must be on Floor 6. Grey is on Floor 4. This is also valid.

4. Let's analyze Case B further:
Floors status so far:
6: Grey
5:
4: P (White)
3: U
2:
1:
Clues for Pink, Blue, and Q:
- "Only one floor is between Q and the one who likes pink."
- "The one who likes Pink and the one who likes blue live on the adjacent floor."
- "Neither U nor P likes blue."
If we assign Pink to U (Floor 3):
Since P (Floor 4) likes White, the adjacent Blue-liking person must be on Floor 2. Since U (Floor 3) likes Pink, Q (having one floor between Q and Pink) can be on Floor 5 or Floor 1.
Let Q be on Floor 5. This leaves Floor 1 and Floor 6 for the remaining persons.

5. Clue: "The number of floors between the one who likes black and S is two less than the number of floors between S and the one who likes red."
Let the distance between Black and S be $d{1}_{}$ and between S and Red be $d{2}_{}$. We have:
$d{1}_{}=d{2}_{}-2$.
If S is on Floor 2 (likes Blue), Q on Floor 5 likes Red, and the person on Floor 1 likes Black:
- Floors between S (Floor 2) and Red (Floor 5) = 2 (Floor 3 and 4). So $d{2}_{}=2$.
- Floors between Black (Floor 1) and S (Floor 2) = 0. So $d{1}_{}=0$.
Since $0=2-2$, this satisfies the condition perfectly.

6. Placing the remaining persons (R and T):
T likes neither grey nor red, so T cannot be on Floor 6 (Grey) or Floor 5 (Red). Thus, T must be on Floor 1 (Black). This leaves R to be placed on Floor 6 (Grey).

The final correct arrangement is:
- Floor 6: R (Grey)
- Floor 5: Q (Red)
- Floor 4: P (White)
- Floor 3: U (Pink)
- Floor 2: S (Blue)
- Floor 1: T (Black)

Evaluating the statements:
- Statement I: T lives above R. (False, T lives on Floor 1 and R lives on Floor 6)
- Statement II: Q lives on an odd numbered floor. (True, Q lives on Floor 5)
- Statement III: R lives on the adjacent floor of Q. (True, R lives on Floor 6 and Q lives on Floor 5, which are adjacent)

Therefore, both Statement II and Statement III are true.