The correct option is Two.

Let us determine the schedule day-by-day from Monday to Sunday based on the given clues:

Step 1: Position D and C
- "Only two people have exams before D." This means D's exam must be on the third day of the week, which is Wednesday.
- "Only one person has exam after C." This means C's exam must be on the second-to-last day of the week, which is Saturday.

Step 2: Position Z
- "Only three people have exams between D and Z."
Since D is on Wednesday, if we go forward, the three days in between are Thursday, Friday, and Saturday. This places Z on Sunday. (We cannot go backward as there are only two days before Wednesday).

So far, our schedule looks like this:
- Monday: _
- Tuesday: _
- Wednesday: D
- Thursday: _
- Friday: _
- Saturday: C
- Sunday: Z

Step 3: Position F, X, Y, and E
- "F has exam immediately before X." This means F and X must occupy consecutive vacant days: (F, X). The available consecutive slots are either (Monday, Tuesday) or (Thursday, Friday).
- "Only one person has exam between X and Y."
Let's test both possibilities for the (F, X) pair:
- Case 1: F is on Thursday and X is on Friday. If X is on Friday, then Y must be on either Wednesday or Sunday (to leave exactly one day between them). However, Wednesday is occupied by D, and Sunday is occupied by Z. Thus, this case is invalid.
- Case 2: F is on Monday and X is on Tuesday. If X is on Tuesday, then Y must be on Thursday to leave exactly one day (Wednesday, occupied by D) between them. This is valid! This leaves Friday as the only vacant slot, which must be occupied by E.

Final Schedule:
- Monday: F
- Tuesday: X
- Wednesday: D
- Thursday: Y
- Friday: E
- Saturday: C
- Sunday: Z

Step 4: Answer the question
We need to find the number of people who have exams between E (Friday) and X (Tuesday).
The people scheduled between Tuesday and Friday are those on Wednesday (D) and Thursday (Y).
Therefore, exactly two people (D and Y) have exams between E and X.