Question

Consider a five-digit number \(PQRST\) that has distinct digits \(P, Q, R, S,\) and \(T\), and satisfies the following conditions: \[ P<Q, S>P>T, R<T \] If integers 1 through 5 are used to construct such a number, the value of \(P\) is:

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

When inequalities place exactly two variables smaller and two larger than one value, that value must be the median of the set.

Answer 1

The Correct Option is C

Step 1: Understand the inequalities.
From the conditions: - \(S>P>T\) means \(P\) is between \(S\) and \(T\). - \(R<T\). - Also, \(P<Q\). So the relative order is: \[ R<T<P<S, Q. \]

Step 2: Identify the position of \(P\).
Since the digits \(\{1,2,3,4,5\}\) must all be used, there must be exactly: - two digits smaller than \(P\) (\(R, T\)), - and two digits larger than \(P\) (\(Q, S\)). Thus, \(P\) is the middle element, i.e., the median of \(\{1,2,3,4,5\}\).

Step 3: Conclusion.
Therefore, \[ P = 3 \] Final Answer: \[ \boxed{3} \]