This problem is a logic puzzle involving the scheduling of seven students (J, K, L, M, N, O, P) admitting into a school over seven days of a week (Monday to Sunday). We need to determine the admission day for student N based on the given clues.

Step-by-Step Deduction of Admission Schedule

Let's break down the information and deduce the schedule:

• Clue 1: J takes admission on Wednesday.
  • Wednesday: J
• Clue 2: Only one student takes admission between J and M.
  • Since J is on Wednesday, M can either be on Monday (M -> X -> J) or Friday (J -> X -> M).
  • Case 1: M is on Monday. The sequence is M (Mon), Tue (X), J (Wed). This fits the clue.
  • Case 2: M is on Friday. The sequence is J (Wed), Thu (X), M (Fri). This also fits the clue.
• Clue 3: L takes admission before M and after K. This establishes the order K → L → M.
  • Consider Case 1 (M is on Monday): If M is on Monday, K and L must be admitted before Monday, which is impossible in a Monday-Sunday week. Therefore, Case 1 is invalid.
  • Consider Case 2 (M is on Friday): M is on Friday. The schedule is: Mon, Tue, J (Wed), Thu, M (Fri), Sat, Sun. Since K → L → M, L must be admitted before Friday, and K must be admitted before L. The available slots before Friday are Monday, Tuesday, and Thursday. L cannot be Thursday because that would imply K must be before Thursday, potentially clashing with J or the structure. Let's try placing L on Tuesday. If L is on Tuesday, K must be on Monday to satisfy K → L.
  • Tentative Schedule based on Clues 1, 2, 3:
    • Monday: K
    • Tuesday: L
    • Wednesday: J
    • Thursday: (Empty)
    • Friday: M
    • Saturday: (Empty)
    • Sunday: (Empty)
• Clue 4: Two students take admission between K and N. K takes admission before N. (K _ _ N)
  • From the tentative schedule, K is admitted on Monday. For two students to be between K and N, N must be admitted on Thursday (K (Mon), Tue, Wed, N (Thu)).
  • The students admitted on Tuesday and Wednesday are L and J, respectively. This fits the condition of two students between K and N.
  • Updated Schedule:
    • Monday: K
    • Tuesday: L
    • Wednesday: J
    • Thursday: N
    • Friday: M
    • Saturday: (Empty)
    • Sunday: (Empty)
• Clue 5: O takes admission in one of the days after M.
  • M is admitted on Friday. The days after Friday are Saturday and Sunday. So, O must be admitted on either Saturday or Sunday.
• Clue 6: P does not take admission on Sunday.
  • The remaining students are O and P. The remaining days are Saturday and Sunday.
  • Since P cannot be admitted on Sunday, P must be admitted on Saturday.
  • Consequently, O must be admitted on Sunday.

Final Admission Schedule

Based on the step-by-step deduction, the complete admission schedule is:

Conclusion for N's Admission Day

According to the derived schedule, student N takes admission on Thursday.

Therefore, the correct option is 3.