Question:
Directions: Each group of questions in this section is based on a set of conditions. Choose the response that most accurately and completely answers each question.
(Questions 108 to 112): Krishnapuram's town council has exactly three members: Arjun, Karn, and Bhim. During one week, the council members vote on exactly three bills: a recreation bill, a school bill, and a tax bill. Each council member votes either for or against each bill. The following is known:
- Each member of the council votes for at least one of the bills and against at least one of the bills.
- Exactly two members of the council vote for the recreation bill.
- Exactly one member of the council votes for the school bill.
- Exactly one member of the council 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.
112. If one of the members of the council votes against exactly the same bills as does another member of the council, then which one of the following statements must be true?
Directions: Each group of questions in this section is based on a set of conditions. Choose the response that most accurately and completely answers each question.
(Questions 108 to 112): Krishnapuram's town council has exactly three members: Arjun, Karn, and Bhim. During one week, the council members vote on exactly three bills: a recreation bill, a school bill, and a tax bill. Each council member votes either for or against each bill. The following is known:
112. If one of the members of the council votes against exactly the same bills as does another member of the council, then which one of the following statements must be true?
(Questions 108 to 112): Krishnapuram's town council has exactly three members: Arjun, Karn, and Bhim. During one week, the council members vote on exactly three bills: a recreation bill, a school bill, and a tax bill. Each council member votes either for or against each bill. The following is known:
- Each member of the council votes for at least one of the bills and against at least one of the bills.
- Exactly two members of the council vote for the recreation bill.
- Exactly one member of the council votes for the school bill.
- Exactly one member of the council 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.
112. If one of the members of the council votes against exactly the same bills as does another member of the council, then which one of the following statements must be true?
Show Hint
Fix Arjun, Karn and Bhim's known votes first, use the recreation-bill count to place Bhim, then test the four ways to place the single for-votes on school and tax to see which one gives two members an identical against-set.
Updated On: Jul 13, 2026
- Arjun votes for the tax bill
- Karn votes for the recreation bill.
- Karn votes against the school bill.
- Bhim votes for exactly one bill.
Show Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
Step 1: Understanding the Question.
Three council members, Arjun, Karn and Bhim, each vote for or against three bills: recreation, school and tax. We know the base voting pattern from the given conditions, and this question adds one more condition: two members vote against exactly the same set of bills. We need to find what must be true under this extra condition.
Step 2: Key Formula or Approach.
Build a for/against grid using the fixed clues first (recreation, school, tax counts, and the direct statements about Arjun, Karn, Bhim). Then work out every way to fill the remaining unknown votes, keep only the ones that also satisfy "each member votes for at least one bill and against at least one bill", and finally keep only the case where two members share the exact same against set.
Step 3: Detailed Explanation.
Recreation bill: exactly 2 vote for it. Arjun votes for it (given) and Karn votes against it (given), so Bhim must be the second for-vote. Recreation: Arjun for, Karn against, Bhim for.
School bill: exactly 1 votes for it. Arjun votes against school (given), so the single for-vote on school comes from Karn or Bhim.
Tax bill: exactly 1 votes for it. Bhim votes against tax (given), so the single for-vote on tax comes from Arjun or Karn.
Now check the "at least one for, at least one against" rule. Arjun already has a for (recreation) and an against (school), so his tax vote is free. Bhim already has a for (recreation) and an against (tax), so his school vote is free. Karn already has an against (recreation), so Karn needs at least one for-vote, meaning Karn must be for the school bill or the tax bill (or both).
There are four ways to place the single for-vote on school and the single for-vote on tax:
(i) School-for = Karn, Tax-for = Arjun: Arjun(F,A,F), Karn(A,F,A), Bhim(F,A,A). Against sets: Arjun{school}, Karn{recreation,tax}, Bhim{school,tax}. All different.
(ii) School-for = Karn, Tax-for = Karn: Arjun(F,A,A), Karn(A,F,F), Bhim(F,A,A). Against sets: Arjun{school,tax}, Karn{recreation}, Bhim{school,tax}. Arjun and Bhim match.
(iii) School-for = Bhim, Tax-for = Arjun: this puts Karn against all three bills, which breaks the "at least one for" rule for Karn, so it is rejected.
(iv) School-for = Bhim, Tax-for = Karn: Arjun(F,A,A), Karn(A,A,F), Bhim(F,F,A). Against sets: Arjun{school,tax}, Karn{recreation,school}, Bhim{tax}. All different.
Only case (ii) has two members with an identical against set (Arjun and Bhim both vote against school and tax), which is exactly the extra condition given in this question. So case (ii) is the arrangement we use: Arjun votes for recreation, against school, against tax. Karn votes against recreation, for school, for tax. Bhim votes for recreation, against school, against tax.
Checking the options against case (ii): "Arjun votes for the tax bill" is false. "Karn votes for the recreation bill" is false (Karn is fixed against recreation). "Karn votes against the school bill" is false (Karn is for school here). "Bhim votes for exactly one bill" is true, since Bhim is for recreation only and against school and tax.
Step 4: Final Answer.
Case (ii) is the only arrangement consistent with the extra condition, and in it Bhim votes for exactly one bill.\[ \boxed{\text{Bhim votes for exactly one bill.}} \]
Three council members, Arjun, Karn and Bhim, each vote for or against three bills: recreation, school and tax. We know the base voting pattern from the given conditions, and this question adds one more condition: two members vote against exactly the same set of bills. We need to find what must be true under this extra condition.
Step 2: Key Formula or Approach.
Build a for/against grid using the fixed clues first (recreation, school, tax counts, and the direct statements about Arjun, Karn, Bhim). Then work out every way to fill the remaining unknown votes, keep only the ones that also satisfy "each member votes for at least one bill and against at least one bill", and finally keep only the case where two members share the exact same against set.
Step 3: Detailed Explanation.
Recreation bill: exactly 2 vote for it. Arjun votes for it (given) and Karn votes against it (given), so Bhim must be the second for-vote. Recreation: Arjun for, Karn against, Bhim for.
School bill: exactly 1 votes for it. Arjun votes against school (given), so the single for-vote on school comes from Karn or Bhim.
Tax bill: exactly 1 votes for it. Bhim votes against tax (given), so the single for-vote on tax comes from Arjun or Karn.
Now check the "at least one for, at least one against" rule. Arjun already has a for (recreation) and an against (school), so his tax vote is free. Bhim already has a for (recreation) and an against (tax), so his school vote is free. Karn already has an against (recreation), so Karn needs at least one for-vote, meaning Karn must be for the school bill or the tax bill (or both).
There are four ways to place the single for-vote on school and the single for-vote on tax:
(i) School-for = Karn, Tax-for = Arjun: Arjun(F,A,F), Karn(A,F,A), Bhim(F,A,A). Against sets: Arjun{school}, Karn{recreation,tax}, Bhim{school,tax}. All different.
(ii) School-for = Karn, Tax-for = Karn: Arjun(F,A,A), Karn(A,F,F), Bhim(F,A,A). Against sets: Arjun{school,tax}, Karn{recreation}, Bhim{school,tax}. Arjun and Bhim match.
(iii) School-for = Bhim, Tax-for = Arjun: this puts Karn against all three bills, which breaks the "at least one for" rule for Karn, so it is rejected.
(iv) School-for = Bhim, Tax-for = Karn: Arjun(F,A,A), Karn(A,A,F), Bhim(F,F,A). Against sets: Arjun{school,tax}, Karn{recreation,school}, Bhim{tax}. All different.
Only case (ii) has two members with an identical against set (Arjun and Bhim both vote against school and tax), which is exactly the extra condition given in this question. So case (ii) is the arrangement we use: Arjun votes for recreation, against school, against tax. Karn votes against recreation, for school, for tax. Bhim votes for recreation, against school, against tax.
Checking the options against case (ii): "Arjun votes for the tax bill" is false. "Karn votes for the recreation bill" is false (Karn is fixed against recreation). "Karn votes against the school bill" is false (Karn is for school here). "Bhim votes for exactly one bill" is true, since Bhim is for recreation only and against school and tax.
Step 4: Final Answer.
Case (ii) is the only arrangement consistent with the extra condition, and in it Bhim votes for exactly one bill.\[ \boxed{\text{Bhim votes for exactly one bill.}} \]
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