Nine persons purchase different products one after the other. Either four or five persons purchase after K. Three persons purchase between K and L. M purchases just after L. Number of persons purchase before M is same as after N. O purchases between N and K. Q purchases just after P but before O. R purchases before S but after O.

Who among the following purchase the product exactly between S and K?

To find who purchases the product exactly between S and K, we can reconstruct the purchasing order step-by-step from the given clues.

Step 1: Understand the basic constraints.
There are 9 persons in total. Let the positions of purchase be numbered 1 to 9, where 1 is the first person to purchase and 9 is the last person.

Step 2: Analyze the position of K.
The problem states that either four or five persons purchase after K.
- If 4 persons purchase after K, K is at position 5 (since 9 - 4 = 5).
- If 5 persons purchase after K, K is at position 4 (since 9 - 5 = 4).

Step 3: Analyze the position of L and M.
Three persons purchase between K and L. Let's analyze both possibilities for K:

Case 1: K is at position 5.
- L can be either at position 1 (three persons between 1 and 5: positions 2, 3, and 4) or position 9 (three persons between 5 and 9: positions 6, 7, and 8).
- We are told: "M purchases just after L".
- If L is at position 9, M cannot purchase after L because there is no position 10. Thus, L must be at position 1, which means M is at position 2.
- Let's check the condition: "Number of persons purchase before M is same as after N".
- M is at position 2, so the number of persons purchasing before M is 1 (position 1).
- Therefore, the number of persons purchasing after N must also be 1. This places N at position 8.
- Now we have: L = 1, M = 2, K = 5, N = 8.

Case 2: K is at position 4.
- L can be at position 8 (three persons between 4 and 8: positions 5, 6, and 7). L cannot be at position 0.
- Since M purchases just after L, M must be at position 9.
- The number of persons purchasing before M is 8 (positions 1 to 8).
- Therefore, the number of persons purchasing after N must be 8, which places N at position 1.
- Now we have: N = 1, K = 4, L = 8, M = 9.

Step 4: Analyze the position of O.
The problem states: "O purchases between N and K".
- In Case 1: K is at 5 and N is at 8. The positions between them are 6 and 7. Thus, O is at 6 or 7.
- In Case 2: N is at 1 and K is at 4. The positions between them are 2 and 3. Thus, O is at 2 or 3.

Step 5: Position of P and Q.
The clue states: "Q purchases just after P but before O". This means P and Q are adjacent (PQ), and both are before O.
- In Case 1: O is at 6 or 7. If O is at 6, the positions before O are 1, 2, 3, 4, 5. Since L=1, M=2, K=5, the only consecutive empty positions left before 6 are 3 and 4. So, P = 3 and Q = 4. Let's test this: if P = 3, Q = 4, K = 5, O = 6, N = 8, then the remaining empty positions are 7 and 9. If O is at 7 instead, the available positions before 7 are 3, 4, 5, 6. Since K = 5, we could have P = 3, Q = 4, and O = 7, leaving position 6 free.
- In Case 2: O is at 2 or 3. If O is at 2 or 3, the positions before O can only be 1 and 2. But N is at 1, so there is no space for the consecutive pair PQ before O. Thus, Case 2 is invalid.

Let's complete Case 1:
Positions: 1 (L), 2 (M), 3 (P), 4 (Q), 5 (K), 6 (O), 7, 8 (N), 9.
We have O at position 6. This satisfies "O purchases between N (8) and K (5)" and "Q (4) purchases just after P (3) but before O (6)".
This leaves positions 7 and 9 vacant.

Step 6: Position of R and S.
The final clue states: "R purchases before S but after O".
- Since O is at position 6, and the remaining vacant positions are 7 and 9, R must be at position 7 and S must be at position 9. This satisfies "R purchases before S (7 before 9) but after O (7 after 6)".

Final Ordering:
1. L
2. M
3. P
4. Q
5. K
6. O
7. R
8. N
9. S

Answering the Question:
We need to find who purchases the product exactly between S (position 9) and K (position 5).
The positions between 5 and 9 are 6, 7, and 8, which are occupied by O, R, and N respectively.
The exact middle position between 5 (K) and 9 (S) is position 7, which is occupied by R.