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?

• A. Arjun and Karn each vote for exactly one bill.
• B. Karn and Bhim each vote for exactly one bill.
• C. Arjun votes for exactly two bills.
• D. Karn votes for the recreation bill.

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.

Answer 1

The Correct Option is A

Filter the three base grids by this question's condition, "Karn is a yes on tax," then just read the for-vote totals per row for whichever grids survive, instead of re-deriving anything from the original rules.

Draw a 3-by-3 grid, one row per member (Arjun, Karn, Bhim) and one column per bill (recreation, school, tax), and fill in only what the passage states directly: Arjun is yes on recreation and no on school; Karn is no on recreation; Bhim is no on tax.

Recreation needs exactly two yes-votes overall. Arjun already supplies one of those two. Karn is a confirmed no, so the second yes on recreation cannot be Karn, which leaves only Bhim. Fill in Bhim as yes on recreation right away, before touching anything else.

School needs exactly one yes-vote overall, and Arjun's cell there is already a no, so the single yes on school belongs to either Karn or Bhim, not both. Tax needs exactly one yes-vote overall, and Bhim's cell there is already a no, so the single yes on tax belongs to either Arjun or Karn, not both.

Every row of the grid needs at least one yes and at least one no. Arjun's row already has a yes (recreation) and a no (school), so Arjun's tax cell can go either way without breaking this rule. Bhim's row already has a yes (recreation) and a no (tax), so Bhim's school cell can go either way too. Karn's row so far has only a no (recreation), so Karn's row needs at least one yes somewhere among school and tax, or the row rule breaks.

1. Grid A: put the school yes on Karn, and the tax yes on Arjun. This forces Karn to no on tax (tax already has its one yes, from Arjun) and Bhim to no on school (school already has its one yes, from Karn). Row totals: Arjun yes on recreation and tax; Karn yes on school only; Bhim yes on recreation only.
2. Grid B: put the school yes on Karn, and the tax yes on Karn as well. This forces Arjun to no on tax and Bhim to no on school. Row totals: Arjun yes on recreation only; Karn yes on school and tax; Bhim yes on recreation only.
3. Grid C: put the school yes on Bhim instead of Karn. Since Karn's row still needs a yes somewhere, and school is now taken by Bhim, Karn's yes has to be on tax, which also forces Arjun to no on tax. Row totals: Arjun yes on recreation only; Karn yes on tax only; Bhim yes on recreation and school.

Every row in all three grids carries at least one yes and one no, and every column total matches the passage's counts, so all three grids are valid, and since the school/tax assignment choices above cover every possibility, no fourth grid exists.

Now apply the filter: Grid A has Arjun as the tax yes-voter (so Karn is a tax no there), which fails this question's condition, so Grid A is filtered out. Grid B has Karn as the tax yes-voter, so Grid B survives. Grid C also has Karn as the tax yes-voter, so Grid C survives too. Two grids remain.

Grid B row totals: Arjun is yes only on recreation (1 yes), Karn is yes on school and tax (2 yes), Bhim is yes only on recreation (1 yes).
Grid C row totals: Arjun is yes only on recreation (1 yes), Karn is yes only on tax (1 yes), Bhim is yes on recreation and school (2 yes).

1. Arjun and Karn each exactly one yes: fails in Grid B (Karn has 2), succeeds in Grid C (Arjun 1, Karn 1). Achievable.
2. Karn and Bhim each exactly one yes: fails in Grid B (Karn has 2), fails in Grid C (Bhim has 2). Not achievable in either surviving grid.
3. Arjun exactly two yes: Arjun is stuck at 1 yes-vote in both Grid B and Grid C. Not achievable.
4. Karn votes for recreation: Karn is a fixed no on recreation in every grid from the very start. Not achievable.

Grid C is the only place any of the five statements comes true, and it makes "Arjun and Karn each vote for exactly one bill" correct. Every other statement fails in both of the surviving grids, so this is the one that could be true.