Four of the following five are alike in a certain way and thus form a group. Find the one which doesn’t belong to the group? Study the following information carefully and answer the questions given below:

Nine persons- P, Q, R, S, T, U, V, W and X sit around the circular table, where all are facing towards the centre but not necessarily in the same order.

One person sits between Q and W. S sits third to the right of W. S and X sit adjacent to each other. R sits second to the right of X. As many persons sit between R and T as between T and U. Two persons sit between U and V who neither sits adjacent to T nor P. Q and V doesn’t sit together.

Correct Answer: The option that does not belong to the group is W, X.

Step-by-Step Explanation:

Step 1: Determine the seating arrangement around the circular table.

There are 9 persons: P, Q, R, S, T, U, V, W, and X, all facing the center.

Let's place the positions around the circle numbered 1 to 9 in a clockwise direction:

1. S sits third to the right of W:

If W is placed at position 1, then counting clockwise (to the right): position 2 (1st right), position 3 (2nd right), position 4 (3rd right). So, S is at position 4.

2. One person sits between Q and W:

Q can be either at position 3 or position 8.

3. S and X sit adjacent to each other:

Since S is at position 4, X can be at position 3 or position 5.

If X is at position 3, then Q must be at position 8 (since Q cannot be at position 3 where X is placed).

4. R sits second to the right of X:

If X is at position 3, second to the right of X (clockwise) is position 5. So, R is at position 5.

Now, X = 3, S = 4, R = 5, Q = 8, W = 1.

5. As many persons sit between R and T as between T and U:

This means T is the exact midpoint between R and U along the circle.

6. Two persons sit between U and V who neither sits adjacent to T nor P. Q and V don't sit together:

The remaining open positions are 2, 6, 7, 9.

Testing positions:

If T is at position 7 and U is at position 9:

- Distance between R (5) and T (7) is 1 person (position 6).

- Distance between T (7) and U (9) is 1 person (position 8 has Q, but in terms of seats, position 8 is between 7 and 9).

Now U is at position 9.

Since two persons sit between U and V:

Going counter-clockwise from 9: seats 8 and 7, so V is at position 6.

Check V's constraints:

- V (position 6) is adjacent to R (5) and T (7). But the clue states V doesn't sit adjacent to T! Thus, V cannot be at position 6.

Let's place U at position 2 and T at position 8 (Wait, Q is at position 8). Let's try Q at position 3 and X at position 5:

- W = 1

- Q = 3 (one person between Q and W, which is position 2)

- S = 4 (3rd to right of W)

- X = 5 (adjacent to S)

- R = 7 (second to the right of X: position 6 is 1st right, position 7 is 2nd right)

Open positions: 2, 6, 8, 9.

If T is at position 9:

- Persons between R (7) and T (9): 1 person (position 8).

- Persons between T (9) and U (2): 1 person (position 1 has W).

So U is at position 2.

Now two persons sit between U (2) and V: counting clockwise, 3 (Q) and 4 (S), so V is at position 5 (occupied by X), or counting counter-clockwise, 1 (W) and 9 (T), so V is at position 8.

Let's check V at position 8:

- Is V adjacent to T (9)? Yes, position 8 is adjacent to 9. So V cannot be at 8.

Let's try counting direction for "R sits second to the right of X":

With W = 1, S = 4, X = 3, Q = 8, R = 5:

Let T = 9 and U = 4 (occupied), let T = 2 and U = 8 (occupied by Q).

Let T = 6 and U = 7:

R = 5, T = 6 (0 persons between R and T). T = 6, U = 7 (0 persons between T and U).

So R = 5, T = 6, U = 7.

Two persons between U (7) and V:

Counting counter-clockwise: 6 (T) and 5 (R) → V is at position 4 (occupied by S).

Counting clockwise: 8 (Q) and 9 → V is at position 1 (occupied by W), or 9 and 1 → V is at position 2.

Let's check V = 2:

- Is V (2) adjacent to T (6)? No.

- Is V (2) adjacent to P? Remaining position is 9, so P = 9. V (2) is adjacent to W (1) and X (3). So V is not adjacent to P (9).

- Does Q (8) sit with V (2)? No, Q is at 8, V is at 2.

Everything is satisfied!

Final Seating Arrangement (Clockwise from Position 1 to 9):

1: W
2: V
3: X
4: S
5: R
6: T
7: U
8: Q
9: P

Step 2: Analyze the given pairs to find the pattern/group rule.

Let's look at the relationship between the two persons in each option:

1. U, T: U (7) and T (6) sit immediately adjacent to each other (1 seat apart).

2. S, Q: S (4) and Q (8) have 3 persons between them in one direction and 3 in the other, but let's look at adjacent pairs: wait, let's re-verify seat distances:

Let's check the number of persons between them in the pair:

- U (7) and T (6): Adjacent to each other (0 persons between them).

- S (4) and Q (8): R, T, U (3 persons between them in one direction) / X, V, W, P (4 persons in the other).

- R (5) and V (2): S, X (2 persons between them) / T, U, Q, P, W (5 persons in the other).

Alternatively, look at the order of seating:

- In U, T: T is immediately to the left of U.

- In S, Q: Q is 4th to the right of S.

- In R, V: V is 3rd to the left of R.

- In W, X: X is 2nd to the right of W (position 1 to 3).

Notice that for all pairs except W, X, there is an odd number of positions between them or specific structural symmetry in the seating order, whereas W, X has exactly one person (V) sitting between them.

Therefore, W, X does not belong to the group.