GATE CS 2015 Set 3 Q.7 (GA Q.7): Analytical and Logical Reasoning

The head of a newly formed government desires to appoint five of the six selected members P, Q, R, S, T, and U to portfolios of Home, Power, Defense, Telecom, and Finance. U does not want any portfolio if S gets one of the five. R wants either Home or Finance or no portfolio. Q says that if S gets either Power or Telecom, then she must get the other one. T insists on a portfolio if P gets one.

Which is the valid distribution of portfolios?

P-Home, Q-Power, R-Defense, S-Telecom, T-Finance

R-Home, S-Power, P-Defense, Q-Telecom, T-Finance

P-Home, Q-Power, T-Defense, S-Telecom, U-Finance

Q-Home, U-Power, T-Defense, R-Telecom, P-Finance

Official answer and solution

Official answer: B
From: Official GATE 2015 answer key

Solution

Formulate each constraint as a propositional condition and verify each option systematically against the rules.

There are 6 members: $\{P, Q, R, S, T, U\}$ to fill 5 portfolios: $\{\text{Home}, \text{Power}, \text{Defense}, \text{Telecom}, \text{Finance}\}$ , meaning exactly one person gets no portfolio.

Constraints:

If $S$ gets a portfolio, $U$ gets none: $S \implies \neg U$ .
$R$ gets either Home or Finance or no portfolio: $R \in \{\text{Home}, \text{Finance}, \emptyset\}$ .
If $S$ gets Power or Telecom, $Q$ must get the other: $(S = \text{Power} \implies Q = \text{Telecom}) \wedge (S = \text{Telecom} \implies Q = \text{Power})$ .
If $P$ gets a portfolio, $T$ must get one: $P \implies T$ .

Testing options:

Option A: $P$ -Home, $Q$ -Power, $R$ -Defense, $S$ -Telecom, $T$ -Finance.
Here $R$ has Defense, violating Constraint 2 ( $R \in \{\text{Home}, \text{Finance}, \emptyset\}$ ). Invalid.
Option B: $R$ $R$ -Home, $S$ $S$ -Power, $P$ $P$ -Defense, $Q$ $Q$ -Telecom, $T$ $T$ -Finance.
- $R$ has Home (satisfies Constraint 2).
- $S$ has Power, $Q$ has Telecom (satisfies Constraint 3).
- $S$ has a portfolio, $U$ gets none (satisfies Constraint 1).
- $P$ gets Defense and $T$ gets Finance (satisfies Constraint 4).
  All constraints are satisfied! Valid.
Option C: $P$ -Home, $Q$ -Power, $T$ -Defense, $S$ -Telecom, $U$ -Finance.
Here both $S$ and $U$ get portfolios, violating Constraint 1. Invalid.
Option D: $Q$ -Home, $U$ -Power, $T$ -Defense, $R$ -Telecom, $P$ -Finance.
Here $R$ gets Telecom, violating Constraint 2. Invalid.

Final answer

Common trap

Overlooking the restriction on R, which strictly limits R to Home, Finance, or no portfolio.

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2015 Computer Science and Information Technology (CS) Session 3
Question: 7 of 65, printed as GA Q.7
Section: General Aptitude

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