Each of P, Q, R, S, T, U and V has an exam on a different day of a week starting from Monday and ending on Sunday of the same week. Q has an exam on Thursday and U has an exam on Saturday. T has an exam immediately before S. P has an exam on one of the days after V. R has an exam immediately after S. How many people have exams between R and P?
Exam Schedule Constraints Analysis
This problem asks us to figure out the order of exams for seven people (P, Q, R, S, T, U, V) scheduled across a week, from Monday to Sunday. We are given specific conditions to follow and need to determine how many people have exams scheduled between the exams of R and P.
Analyzing the Given Constraints
First, let's list and understand all the rules provided:
- Days Available: Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday.
- People Involved: P, Q, R, S, T, U, V (one person per day).
- Fixed Schedules:
- Q's exam is on Thursday.
- U's exam is on Saturday.
- Consecutive Schedules:
- T's exam is immediately before S's exam. Let's represent this as a 'TS' block.
- R's exam is immediately after S's exam. Let's represent this as an 'SR' block.
- Relative Order: P's exam occurs on a day after V's exam. This means V must come sometime before P in the schedule (V ... P).
Deducing the TSR Block Placement
We have identified a 'TSR' block which requires three consecutive days. Let's see where this block can fit into the week, considering the fixed positions of Q (Thursday) and U (Saturday).
The weekly schedule looks like this:
| Monday | Tuesday | Wednesday | Thursday | Friday | Saturday | Sunday |
| Q | U |
Now, let's evaluate possible slots for the 3-day TSR block:
- Slot 1: Monday - Wednesday If TSR is Mon-Wed, the schedule starts: T S R Q _ U _. This looks possible.
- Slot 2: Tuesday - Thursday This slot is impossible because Q is already scheduled for Thursday. The TSR block cannot occupy Thursday.
- Slot 3: Wednesday - Friday This slot is also impossible because Q is scheduled for Thursday, which falls within this period.
- Slot 4: Thursday - Saturday This is impossible because Q is on Thursday and U is on Saturday, conflicting with the TSR block.
- Slot 5: Friday - Sunday This slot is impossible because U is scheduled for Saturday. If TSR were Fri-Sun, S would have to be on Saturday, conflicting with U.
Based on this analysis, the only possible placement for the TSR block is Monday to Wednesday.
Determining the Final Exam Schedule
With the TSR block confirmed for Monday-Wednesday, and knowing Q is on Thursday and U is on Saturday, we can update the schedule:
- Monday: T
- Tuesday: S
- Wednesday: R
- Thursday: Q
- Friday: (Empty)
- Saturday: U
- Sunday: (Empty)
The people left to schedule are P and V. The remaining days are Friday and Sunday.
We must apply the constraint that P's exam is after V's exam (V ... P).
To satisfy this, V must take the earlier available day (Friday), and P must take the later available day (Sunday).
The complete exam schedule is:
| Monday | Tuesday | Wednesday | Thursday | Friday | Saturday | Sunday |
| T | S | R | Q | V | U | P |
Let's quickly verify this schedule against all the initial constraints:
- Q on Thursday? Yes.
- U on Saturday? Yes.
- T immediately before S? Yes (Mon, Tue).
- R immediately after S? Yes (Tue, Wed).
- P after V? Yes (V on Fri, P on Sun).
- All days and people used? Yes.
The schedule is correct.
Counting People Between R and P
The final step is to count how many people have exams scheduled strictly between R and P.
- R's exam is on Wednesday.
- P's exam is on Sunday.
- The days between Wednesday and Sunday are Thursday, Friday, and Saturday.
- Looking at the final schedule, the people taking exams on these intermediate days are:
- Thursday: Q
- Friday: V
- Saturday: U
- There are exactly 3 people (Q, V, and U) with exams scheduled between R and P.
Important Questions from Ordering and Ranking (Notes)
- The sixth largest three-digit odd number among those with all digits distinct is
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