M has an exam on Thursday. Exactly 4 people have their exams between O and A, neither of whom has an exam on Sunday. Q has an exam immediately before C. I does not have an exam on Friday. O does not have an exam after C.
How many people have exams after A?
The problem requires determining the number of people scheduled for exams after person A, given specific constraints for seven individuals (A, C, I, M, O, Q, W) across a Monday-to-Sunday week.
1. Analyzing Exam Schedule Constraints
- Week Structure: Exams are held from Monday to Sunday.
- Participants: A, C, I, M, O, Q, W.
- Constraint Summary:
- M exams on Thursday.
- Exactly 4 people between O and A.
- O and A exams not on Sunday.
- Q's exam is immediately before C's (QC).
- I's exam is not on Friday.
- O's exam is not after C's (O before C).
2. Determining O and A Placement
The constraint of exactly 4 people between O and A implies their exams must be scheduled 5 days apart. In a 7-day week (Monday to Sunday), the only pair of days 5 days apart is Monday and Saturday. This pairing respects the constraint that neither O nor A can be on Sunday.
Thus, the set of exam days for {O, A} must be {Monday, Saturday}.
3. Placing M and the QC Block
M's exam is fixed on Thursday.
The constraints 'O before C' and 'Q immediately before C' dictate that the QC block must occur sometime after O.
We examine two possible scenarios for the {O, A} placement:
Case 1: O on Monday, A on Saturday
The initial schedule framework is:
- Mon: O
- Tue: ?
- Wed: ?
- Thu: M
- Fri: ?
- Sat: A
- Sun: ?
The QC block needs to fit after O (Monday). Potential slots are (Tue, Wed), (Wed, Thu), or (Thu, Fri). Since M is scheduled on Thursday, the slots (Wed, Thu) and (Thu, Fri) are unavailable for QC.
Therefore, the QC block must occupy Tuesday and Wednesday.
The schedule progresses to: O, Q, C, M, ?, A, ?
The remaining people are I and W, and the remaining days are Friday and Sunday.
According to Constraint 5, I cannot have an exam on Friday. This forces I to be scheduled on Sunday, leaving Friday for W.
This scenario results in the following complete schedule:
| Day | Mon | Tue | Wed | Thu | Fri | Sat | Sun |
| Person | O | Q | C | M | W | A | I |
This arrangement satisfies all the given conditions.
Case 2: A on Monday, O on Saturday
The initial schedule framework is:
- Mon: A
- Tue: ?
- Wed: ?
- Thu: M
- Fri: ?
- Sat: O
- Sun: ?
Constraint 'O before C' requires C's exam to be after O (Saturday). The only available day is Sunday.
The schedule becomes: A, ?, ?, M, ?, O, C
Constraint 'Q immediately before C' implies Q must take the slot immediately preceding Sunday, which is Saturday. However, O is already scheduled for Saturday.
This conflict means Case 2 is not possible.
4. Final Answer Derivation
The only valid exam schedule determined from the constraints is:
- Monday: O
- Tuesday: Q
- Wednesday: C
- Thursday: M
- Friday: W
- Saturday: A
- Sunday: I
A's exam is scheduled for Saturday. The question asks for the number of people with exams scheduled *after* A.
Only person I has an exam on Sunday, which occurs after Saturday.
Therefore, exactly 1 person has an exam after A.
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