Question:
Directions: Each group of questions is based on a set of conditions. Read the passage below and answer the question that follows.
Krishnapuram's town council has exactly three members: Arjun, Karn and Bhim. During one week the council votes on exactly three bills: a recreation bill, a school bill and a tax bill. Each member votes either for or against each bill. The following is known:
- Each member votes for at least one of the bills and against at least one of the bills.
- Exactly two members vote for the recreation bill.
- Exactly one member votes for the school bill.
- Exactly one member votes for the tax bill.
- Arjun votes for the recreation bill and against the school bill.
- Karn votes against the recreation bill.
- Bhim votes against the tax bill.
If Karn votes for the tax bill, then which one of the following statements could be true?
Directions: Each group of questions is based on a set of conditions. Read the passage below and answer the question that follows.
Krishnapuram's town council has exactly three members: Arjun, Karn and Bhim. During one week the council votes on exactly three bills: a recreation bill, a school bill and a tax bill. Each member votes either for or against each bill. The following is known:
If Karn votes for the tax bill, then which one of the following statements could be true?
Krishnapuram's town council has exactly three members: Arjun, Karn and Bhim. During one week the council votes on exactly three bills: a recreation bill, a school bill and a tax bill. Each member votes either for or against each bill. The following is known:
- Each member votes for at least one of the bills and against at least one of the bills.
- Exactly two members vote for the recreation bill.
- Exactly one member votes for the school bill.
- Exactly one member votes for the tax bill.
- Arjun votes for the recreation bill and against the school bill.
- Karn votes against the recreation bill.
- Bhim votes against the tax bill.
If Karn votes for the tax bill, then which one of the following statements could be true?
Show Hint
Karn being for-tax rules out one of the three base worlds (the one where Arjun is the tax yes-voter), leaving two. Count for-votes per member in the remaining two worlds and match against the choices.
Updated On: Jul 13, 2026
- Arjun and Karn each vote for exactly one bill.
- Karn and Bhim each vote for exactly one bill.
- Arjun votes for exactly two bills.
- Karn votes for the recreation bill.
Show Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
Step 1: Force Bhim's recreation vote.
Arjun votes for the recreation bill (given), and exactly two members vote for it. Karn votes against the recreation bill (given), so Karn is not the second yes-vote. The only member left is Bhim, so Bhim must vote for the recreation bill. This is fixed in every valid case.
Step 2: Note who is still free.
So far: Arjun = for-recreation, against-school; Karn = against-recreation; Bhim = for-recreation, against-tax (given). Exactly one member votes for the school bill, and Arjun is against it, so the sole yes-vote on school is Karn or Bhim. Exactly one member votes for the tax bill, and Bhim is against it, so the sole yes-vote on tax is Arjun or Karn.
Step 3: Use the "at least one for, at least one against" rule to pin Karn.
Arjun already has a for-vote (recreation) and an against-vote (school), so Arjun's tax vote is unrestricted by this rule alone. Bhim already has a for-vote (recreation) and an against-vote (tax), so Bhim's school vote is unrestricted by this rule alone. Karn, though, is against on recreation only so far, so Karn needs at least one for-vote among school and tax, otherwise Karn would be against on all three bills, which breaks the rule.
Step 4: Branch on who is the sole yes-vote for school.
Branch 1, Karn is for-school: Karn's need for a for-vote is already met, so Karn's tax vote is free, giving two further options for who is the sole yes-vote on tax, Arjun or Karn.
Branch 1a: Arjun is for-tax. Then Karn is against-tax (tax has only one yes-voter) and Bhim is against-school (school has only one yes-voter). This gives: Arjun(for R, against S, for T), Karn(against R, for S, against T), Bhim(for R, against S, against T).
Branch 1b: Karn is for-tax. Then Arjun is against-tax and Bhim is against-school. This gives: Arjun(for R, against S, against T), Karn(against R, for S, for T), Bhim(for R, against S, against T).
Branch 2, Bhim is for-school: Then Karn is NOT the school yes-voter, so Karn's only way to satisfy "at least one for-vote" is to be the sole yes-voter on tax, forcing Arjun against-tax. This gives: Arjun(for R, against S, against T), Karn(against R, against S, for T), Bhim(for R, for S, against T).
Step 5: Confirm each branch is internally consistent, giving exactly three valid worlds.
World 1 (Branch 1a): Arjun for R,T / against S. Karn for S / against R,T. Bhim for R / against S,T.
World 2 (Branch 1b): Arjun for R / against S,T. Karn for S,T / against R. Bhim for R / against S,T.
World 3 (Branch 2): Arjun for R / against S,T. Karn for T / against R,S. Bhim for R,S / against T.
Every member has at least one for-vote and one against-vote in all three worlds, and every bill's for-count matches the rules, so all three are valid and no other world is possible.
Step 6: Apply "Karn votes for the tax bill" to the three worlds.
World 1 has Arjun as the tax yes-voter, not Karn, so World 1 is ruled out.
World 2 has Karn as the tax yes-voter, so World 2 survives.
World 3 has Karn as the tax yes-voter too, so World 3 survives.
Two worlds remain: World 2 (Arjun for R only; Karn for S and T; Bhim for R only) and World 3 (Arjun for R only; Karn for T only; Bhim for R and S).
Step 7: Count for-votes in each surviving world and test the choices.
World 2: Arjun has 1 for-vote (R), Karn has 2 for-votes (S, T), Bhim has 1 for-vote (R).
World 3: Arjun has 1 for-vote (R), Karn has 1 for-vote (T), Bhim has 2 for-votes (R, S).
"Arjun and Karn each vote for exactly one bill": in World 3, Arjun has 1 and Karn has 1, both match. This works in World 3.
"Karn and Bhim each vote for exactly one bill": in World 2, Karn has 2, not 1; in World 3, Bhim has 2, not 1. Neither world satisfies both at once.
"Arjun votes for exactly two bills": in both surviving worlds Arjun has only 1 for-vote, never 2.
"Karn votes for the recreation bill": Karn is against recreation in every world, including both survivors.
Final Answer:
Only "Arjun and Karn each vote for exactly one bill" is achievable, and it happens in World 3. That is the statement that could be true. \[ \boxed{\text{Arjun and Karn each vote for exactly one bill}} \]
Arjun votes for the recreation bill (given), and exactly two members vote for it. Karn votes against the recreation bill (given), so Karn is not the second yes-vote. The only member left is Bhim, so Bhim must vote for the recreation bill. This is fixed in every valid case.
Step 2: Note who is still free.
So far: Arjun = for-recreation, against-school; Karn = against-recreation; Bhim = for-recreation, against-tax (given). Exactly one member votes for the school bill, and Arjun is against it, so the sole yes-vote on school is Karn or Bhim. Exactly one member votes for the tax bill, and Bhim is against it, so the sole yes-vote on tax is Arjun or Karn.
Step 3: Use the "at least one for, at least one against" rule to pin Karn.
Arjun already has a for-vote (recreation) and an against-vote (school), so Arjun's tax vote is unrestricted by this rule alone. Bhim already has a for-vote (recreation) and an against-vote (tax), so Bhim's school vote is unrestricted by this rule alone. Karn, though, is against on recreation only so far, so Karn needs at least one for-vote among school and tax, otherwise Karn would be against on all three bills, which breaks the rule.
Step 4: Branch on who is the sole yes-vote for school.
Branch 1, Karn is for-school: Karn's need for a for-vote is already met, so Karn's tax vote is free, giving two further options for who is the sole yes-vote on tax, Arjun or Karn.
Branch 1a: Arjun is for-tax. Then Karn is against-tax (tax has only one yes-voter) and Bhim is against-school (school has only one yes-voter). This gives: Arjun(for R, against S, for T), Karn(against R, for S, against T), Bhim(for R, against S, against T).
Branch 1b: Karn is for-tax. Then Arjun is against-tax and Bhim is against-school. This gives: Arjun(for R, against S, against T), Karn(against R, for S, for T), Bhim(for R, against S, against T).
Branch 2, Bhim is for-school: Then Karn is NOT the school yes-voter, so Karn's only way to satisfy "at least one for-vote" is to be the sole yes-voter on tax, forcing Arjun against-tax. This gives: Arjun(for R, against S, against T), Karn(against R, against S, for T), Bhim(for R, for S, against T).
Step 5: Confirm each branch is internally consistent, giving exactly three valid worlds.
World 1 (Branch 1a): Arjun for R,T / against S. Karn for S / against R,T. Bhim for R / against S,T.
World 2 (Branch 1b): Arjun for R / against S,T. Karn for S,T / against R. Bhim for R / against S,T.
World 3 (Branch 2): Arjun for R / against S,T. Karn for T / against R,S. Bhim for R,S / against T.
Every member has at least one for-vote and one against-vote in all three worlds, and every bill's for-count matches the rules, so all three are valid and no other world is possible.
Step 6: Apply "Karn votes for the tax bill" to the three worlds.
World 1 has Arjun as the tax yes-voter, not Karn, so World 1 is ruled out.
World 2 has Karn as the tax yes-voter, so World 2 survives.
World 3 has Karn as the tax yes-voter too, so World 3 survives.
Two worlds remain: World 2 (Arjun for R only; Karn for S and T; Bhim for R only) and World 3 (Arjun for R only; Karn for T only; Bhim for R and S).
Step 7: Count for-votes in each surviving world and test the choices.
World 2: Arjun has 1 for-vote (R), Karn has 2 for-votes (S, T), Bhim has 1 for-vote (R).
World 3: Arjun has 1 for-vote (R), Karn has 1 for-vote (T), Bhim has 2 for-votes (R, S).
"Arjun and Karn each vote for exactly one bill": in World 3, Arjun has 1 and Karn has 1, both match. This works in World 3.
"Karn and Bhim each vote for exactly one bill": in World 2, Karn has 2, not 1; in World 3, Bhim has 2, not 1. Neither world satisfies both at once.
"Arjun votes for exactly two bills": in both surviving worlds Arjun has only 1 for-vote, never 2.
"Karn votes for the recreation bill": Karn is against recreation in every world, including both survivors.
Final Answer:
Only "Arjun and Karn each vote for exactly one bill" is achievable, and it happens in World 3. That is the statement that could be true. \[ \boxed{\text{Arjun and Karn each vote for exactly one bill}} \]
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