This problem requires us to determine the order of exams for seven people (P, Q, R, S, T, U, V) over a week (Monday to Sunday) based on several given conditions and then find the number of people whose exams fall between V's and T's exams.

Understanding the Constraints

• There are 7 people: P, Q, R, S, T, U, V.
• There are 7 days: Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday. Each person has an exam on a different day.
• Condition 1: Five people have exams between T and Q. This means T and Q must be separated by exactly 5 other people. In a 7-day week sequence, this is only possible if T and Q are on the first and last day, or vice versa. The sequence would look like T _ _ _ _ _ Q or Q _ _ _ _ _ T.
• Condition 2: U has an exam on Thursday. This gives us a fixed point in the schedule.
• Condition 3: S has an exam immediately after Q. This means Q and S form a block (Q, S).
• Condition 4: R has an exam immediately after P. This means P and R form a block (P, R).

Deducing the Schedule

Let's use the conditions to build the weekly schedule.

Applying the T-Q Constraint

From Condition 1, T and Q must be at the ends of the week (Monday and Sunday) to have 5 people between them.

• Scenario 1: T is on Monday, Q is on Sunday. The sequence is T _ _ _ _ _ Q.
• Scenario 2: Q is on Monday, T is on Sunday. The sequence is Q _ _ _ _ _ T.

Evaluating Scenario 1 (T=Mon, Q=Sun)

If Q is on Sunday, S must be immediately after Q (Condition 3). However, there is no day after Sunday in the same week. Therefore, this scenario is impossible.

Evaluating Scenario 2 (Q=Mon, T=Sun)

This scenario must be the correct one. Q is on Monday and T is on Sunday.

Applying Condition 3 (S is immediately after Q): Q is on Monday, so S must be on Tuesday.

Applying Condition 2 (U is on Thursday): U is on Thursday.

Let's mark the known positions:

Mon: Q

Tue: S

Wed: ?

Thu: U

Fri: ?

Sat: ?

Sun: T

The sequence so far is: Q (Mon), S (Tue), _ (Wed), U (Thu), _ (Fri), _ (Sat), T (Sun).

Placing the Remaining People (P, R, V)

The remaining people are P, R, and V. The remaining days are Wednesday, Friday, and Saturday.

We must apply Condition 4: R is immediately after P (the block P, R).

Let's see where the (P, R) block can fit in the remaining days (Wed, Fri, Sat):

• If P is on Wednesday, R would need to be on Thursday. But Thursday is already occupied by U. So, P cannot be on Wednesday.
• If P is on Friday, R must be on Saturday. This fits perfectly, as Friday and Saturday are available and consecutive.

This placement means P is on Friday and R is on Saturday.

The schedule now is:

Mon: Q

Tue: S

Wed: ?

Thu: U

Fri: P

Sat: R

Sun: T

The only remaining person is V, and the only remaining day is Wednesday. Therefore, V must have an exam on Wednesday.

The Final Schedule

The complete schedule is:

Calculating People Between V and T

Now we need to find how many people have exams between V and T.

• V has an exam on Wednesday.
• T has an exam on Sunday.

The days between Wednesday and Sunday are Thursday, Friday, and Saturday.

The people scheduled on these days are:

• Thursday: U
• Friday: P
• Saturday: R

There are 3 people (U, P, and R) who have exams between V and T.