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 purchases the product four persons after M?

To find the correct ordering of the nine persons purchasing the products, let us analyze the given conditions step-by-step:

Step 1: Determine the position of K
Either four or five persons purchase after K. Since there are 9 persons in total, the positions are numbered 1 to 9 from first to last.
- If 4 persons purchase after K, K is at position: 9 - 4 = 5.
- If 5 persons purchase after K, K is at position: 9 - 5 = 4.
Thus, K is at either position 4 or 5.

Step 2: Determine the position of L and M
We are given that three persons purchase between K and L, and M purchases just after L.
Let us evaluate the two cases for K:
- Case 1: K is at position 5
L can be at position 1 (since 5 - 3 - 1 = 1) or position 9 (since 5 + 3 + 1 = 9).
If L is at 9, M (who is just after L) would be at 10, which is impossible since there are only 9 people. Thus, L must be at position 1, and M is at position 2.
- Case 2: K is at position 4
L must be at position 8 (since 4 + 3 + 1 = 8). Consequently, M is at position 9.

Step 3: Analyze the position of N
The number of persons purchasing before M is equal to the number of persons purchasing after N.
- In Case 1: M is at 2. The number of persons before M is 1. Thus, there must be 1 person after N, meaning N is at position 8.
- In Case 2: M is at 9. The number of persons before M is 8. Thus, there must be 8 persons after N, meaning N is at position 1.

Step 4: Use the remaining conditions to find the unique arrangement
We are told that O purchases between N and K, and Q purchases just after P but before O. Also, R purchases before S but after O.
- Let us test Case 2 (N = 1, K = 4):
O must be between N (1) and K (4), so O can only be at position 2 or 3. But P and Q must both purchase before O, with Q just after P (P, Q consecutive). There are not enough vacant positions before position 3 to place P and Q. Thus, Case 2 is invalid.
- Let us test Case 1 (L = 1, M = 2, K = 5, N = 8):
O must be between K (5) and N (8), so O can be at position 6 or 7.
P and Q must purchase before O. Since positions 3 and 4 are vacant, we can place P at 3 and Q at 4. This places both of them before O.
If O is at position 6, the remaining vacant positions are 7 and 9. Since R purchases before S but after O, R must be at position 7 and S must be at position 9. This satisfies all conditions perfectly.

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

Conclusion:
We need to find who purchases the product four persons after M.
Since M is at position 2, the person who purchases four persons after M is at position:
$2\; +\; 4\; =\; 6$
The person at position 6 is O.

Therefore, the correct option is O.