Question

Six students P, Q, R, S, T and U, with distinct heights, compare their heights and make the following observations.
Observation I: S is taller than R.
Observation II: Q is the shortest of all.
Observation III: U is taller than only one student.
Observation IV: T is taller than S but is not the tallest.
The number of students that are taller than R is the same as the number of students shorter than \(\underline{\hspace{2cm}.}\)

• A. T
• B. R
• C. S
• D. P

Hint:

In ordering problems, first fix extreme positions (shortest/tallest), then place "second shortest/second tallest" conditions, and finally count ranks to match the required numbers.

Answer 1

The Correct Option is C

Step 1: Place Q and U using Observations II and III.
From Observation II, \(Q\) is the shortest.
From Observation III, \(U\) is taller than only one student, so \(U\) must be the second shortest.
Hence: \[ Q < U < \text{(remaining four: P, R, S, T)} \]

Step 2: Use Observations I and IV to order R, S, T.
From Observation I: \[ S > R \] From Observation IV: \[ T > S \text{and} T \text{ is not the tallest} \] So among \(R, S, T\), we must have: \[ R < S < T \]

Step 3: Identify who is the tallest.
Since \(T\) is not the tallest, someone must be taller than \(T\).
The only remaining person is \(P\). Hence: \[ P > T \] So the complete order (shortest to tallest) becomes: \[ Q < U < R < S < T < P \]

Step 4: Count students taller than R.
Students taller than \(R\) are: \[ S,\, T,\, P \] So, number of students taller than \(R\) \(= 3\).

Step 5: Find who has exactly 3 students shorter than them.
From the order: \[ Q(1),\ U(2),\ R(3),\ S(4),\ T(5),\ P(6) \] Students shorter than \(S\) are: \[ Q,\, U,\, R \] So, number of students shorter than \(S\) \(= 3\).

Step 6: Conclusion.
Number taller than \(R\) \(= 3\) equals number shorter than \(S\) \(= 3\).
Therefore, the blank must be \(S\).

Final Answer: (C) S