Understanding the Portfolio Assignment Problem

The question presents a scenario where a newly formed government needs to assign five specific portfolios (Home, Power, Defence, Telecom, Finance) to five out of six selected members (P, Q, R, S, T, and U). The challenge lies in satisfying a set of conditional constraints provided by the members themselves. Our goal is to identify the single valid distribution of portfolios from the given options that adheres to all stated conditions.

Analyzing Individual Constraints for Portfolio Distribution

Let's break down each condition related to the portfolio distribution:

• Constraint 1: U does not want any portfolio if S gets one of the five.
  • This implies a direct conflict: if S is assigned a portfolio, then U cannot be assigned a portfolio.
  • Conversely, if U is assigned a portfolio, then S cannot be assigned a portfolio.
  • Since only five portfolios are available for six members, exactly one member will not receive a portfolio. This constraint specifies that if S is among the five assigned, U must be the one left out, or vice-versa.
• Constraint 2: R wants either Home or Finance or no portfolio.
  • This means R's assigned portfolio must be either Home or Finance.
  • Alternatively, R could be the member who does not receive a portfolio.
  • R cannot be assigned Defence, Power, or Telecom.
• Constraint 3: Q says that if S gets either Power or Telecom, then she must get the other one.
  • This is a conditional relationship between Q and S's portfolios.
  • If S gets Power, then Q must get Telecom.
  • If S gets Telecom, then Q must get Power.
• Constraint 4: T insists on a portfolio if P gets one.
  • This is another conditional statement: If P is assigned a portfolio, then T must also be assigned a portfolio.
  • In other words, it is not possible for P to have a portfolio while T does not.
  • If T is the member without a portfolio, then P must also be without a portfolio.

Evaluating Portfolio Distribution Options

Now, let's examine each given option against these four constraints to find the valid distribution.

Option 1: P – Home, Q – Power, R – Defence, S – Telecom, T – Finance

• Constraint 1 (U/S): S gets Telecom. U is not listed, implying U is the unassigned member. This satisfies the constraint.
• Constraint 2 (R): R gets Defence. This violates Constraint 2, which states R wants either Home or Finance or no portfolio.
• Conclusion for Option 1: Invalid.

Option 2: R – Home, S – Power, P – Defence, Q – Telecom, T – Finance

• Constraint 1 (U/S): S gets Power. U is not listed, implying U is the unassigned member. This satisfies the constraint.
• Constraint 2 (R): R gets Home. This satisfies Constraint 2, as Home is one of R's preferred portfolios.
• Constraint 3 (Q/S): S gets Power. The condition requires Q to get Telecom. In this option, Q gets Telecom. This satisfies the constraint.
• Constraint 4 (T/P): P gets Defence. T gets Finance. Both P and T are assigned portfolios. This satisfies the constraint.
• Conclusion for Option 2: Valid. This option satisfies all conditions.

Option 3: P – Home, Q – Power, T – Defence, S – Telecom, U – Finance

• Constraint 1 (U/S): S gets Telecom AND U gets Finance. This directly violates Constraint 1, which states U does not want any portfolio if S gets one.
• Conclusion for Option 3: Invalid.

Option 4: Q – Home, U – Power, T – Defence, R – Telecom, P – Finance

• Constraint 1 (U/S): U gets Power. S is not listed, implying S is the unassigned member. This satisfies the constraint (since U is assigned, S is not).
• Constraint 2 (R): R gets Telecom. This violates Constraint 2, as R only wants Home or Finance or no portfolio.
• Conclusion for Option 4: Invalid.

Final Valid Distribution of Portfolios

Based on the thorough evaluation of all options against the given constraints, only Option 2 adheres to all the specified conditions. Therefore, the valid distribution of portfolios is:

In this distribution, U is the member who does not receive a portfolio, which is consistent with the conditions.