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

Nine persons K, J, Y, R, D, L, X, P and S sit around a circular table but not necessarily in the same order. All of them are facing away from the centre of the table. K sits 3rd to the left of J. One person sits between K and X. Two persons sit between K and L. D sits immediately right of R and both of them are not adjacent to L. At least three persons sit between P and Y (both from left or right) . Y is not adjacent to S.

What will be the difference between the number of persons sit between P and Y when counted from the left of Y and the number of persons sit between D and L when counted from the left of D?

Correct Answer: Option 1

Step-by-step Explanation and Seating Arrangement:

Let's analyze the given conditions step-by-step to arrange the 9 persons (K, J, Y, R, D, L, X, P, and S) around a circular table, all facing away from the centre. Note that when facing away from the centre, clockwise direction represents left, and counter-clockwise direction represents right.

1. Position K and J:
- K sits 3rd to the left of J.
- Let J be at position 1. Since J faces away from the centre, left of J is clockwise.
- Position of K = 1 + 3 = Position 4 (clockwise).

2. Position X:
- One person sits between K and X.
- X can be at position 2 or position 6.
- Let's analyze Case 1 (X at position 2) and Case 2 (X at position 6).

3. Position L:
- Two persons sit between K and L.
- K is at position 4. Moving 3 steps counter-clockwise gives position 1 (occupied by J). Moving 3 steps clockwise gives position 7.
- So, L must sit at position 7.

4. Position D and R:
- D sits immediately right of R.
- Facing away from the centre, "immediately right" means one step counter-clockwise.
- Both D and R are not adjacent to L (position 7), so neither D nor R can be at position 6 or position 8.
- In Case 1 (X at position 2), the vacant positions are 3, 5, 6, 8, 9. Since positions 6 and 8 are excluded for D and R, the continuous available pair (R, D in counter-clockwise order) can fit at positions 9 and 8 (invalid as D at 8) or positions 8 and 7 (invalid) or positions 5 and 4 (invalid). Thus, R must be at position 9 and D at position 8 (Wait: counter-clockwise right means if R is at 9, right of R is 8 in CCW direction. Let's check: 9 -> 8 is CCW, which is right). But D cannot be adjacent to L (7). So D cannot be at 8.
- Therefore, the remaining valid positions for (R, D) counter-clockwise order are positions 3 and 2 or 5 and 4, etc. In Case 2 (X at position 6), positions 6 and 7 are occupied. Available positions are 2, 3, 5, 8, 9. D cannot be at 6 or 8. Thus, R is at position 3 and D is at position 2 (since 3 to 2 is counter-clockwise/right).

5. Position P, Y, and S:
- With X at 6, L at 7, D at 2, R at 3, K at 4, J at 1:
- Remaining positions are 5, 8, 9 for P, Y, and S.
- At least three persons sit between P and Y (both from left or right). Around a 9-person table, if at least 3 persons sit between two people from both sides, the gap must be 3 on one side and 4 on the other side.
- Checking positions (5, 9): gap between 5 and 9 is 3 persons (6, 7, 8) on one side and 4 persons (1, 2, 3, 4) on the other side. This satisfies the condition!
- Thus, {P, Y} occupy positions 5 and 9, and S occupies position 8.
- Since Y is not adjacent to S (position 8), Y cannot be at position 9. Therefore, Y is at position 5, and P is at position 9.

Final Seating Arrangement (Clockwise from position 1 to 9):
- Position 1: J
- Position 2: D
- Position 3: R
- Position 4: K
- Position 5: Y
- Position 6: X
- Position 7: L
- Position 8: S
- Position 9: P

Calculating the required difference:
- Number of persons sitting between P and Y when counted from the left of Y (clockwise from position 5 to position 9):
Between 5 and 9 in clockwise order are positions 6, 7, 8 (X, L, S), which gives 3 persons.
- Number of persons sitting between D and L when counted from the left of D (clockwise from position 2 to position 7):
Between 2 and 7 in clockwise order are positions 3, 4, 5, 6 (R, K, Y, X), which gives 4 persons.
- Difference = $4-3=1$

Therefore, the difference between the two counts is 1.