Question

Which two candidates can belong to the same department?

• A. Prof. Pakrasi and Prof. Qureshi
• B. Prof. Qureshi and Prof. Ramaswamy
• C. Prof. Pakrasi and Prof. Samuel
• D. Prof. Ramaswamy and Prof. Samuel
• A. 7
• B. 8
• C. 6
• D. 9
• A. Only statement A
• B. Both statements A and B
• C. Neither statement A nor statement B
• D. Only statement B
• A. It was either Prof. Ramaswamy or Prof. Samuel.
• B. It was Prof. Samuel.
• C. It was either Prof. Pakrasi or Prof. Qureshi.
• D. It was Prof. Ramaswamy.
• A. Neither statement A nor statement B
• B. Both statements A and B
• C. Only statement A
• D. Only statement B

Answer 1

The Correct Option is A

Provided Information:

• Only 1 candidate originates from O&Q, resulting in 4 non-candidates within O&Q.
• Departments vote as a cohesive unit.
• A minimum of 1 non-candidate vote is required from B&H if 2 of its 3 members are candidates.
• Prof. R secured 5 votes from non-candidates.

Potential Vote Distributions for 5 Non-Candidate Votes:

• (a) A single department provided all 5 votes.
• (b) 4 votes from one department and 1 from another. This scenario implies O&Q provided 4 votes, necessitating 2 candidates in B&H for the remaining 1 vote.
• (c) 3 votes from one department and 2 from another. This scenario also implies 2 candidates in smaller departments.

Analysis of Scenarios:

• Cases (b) and (c) are deemed invalid because they contradict the constraint of only 1 candidate from O&Q and complicate balancing votes across other departments.
• Case (a) is only feasible if all 5 non-candidate votes originated from a single department. This is exclusively possible within M&S, which has 7 total members, 2 of whom are candidates, leaving 5 non-candidate voters.

Resulting Departmental Breakdown:

	F&A	M&S	O&Q	B&H
Total members	9	7	5	3
Candidates	0	2	1	1
Non-candidates	9	5	4	2

Distribution of Votes by Candidate:

	P	Q	R	S
Total Votes	3	14	6	1
Self Vote	1 (S)	1 (R)	1 (P)	1 (Q)
From Non-candidates	2	13	5	0
Source Dept(s)	B&H	F&A + O&Q	M&S	—

Deductions from M&S:
M&S has 2 candidates. Given that Prof. R received 5 votes from M&S non-candidates, R is not a candidate from M&S (as his department voted for him). The potential candidate pairs from M&S are therefore (P, Q), (P, S), or (Q, S).

Considering the vote data and departmental constraints, the pair P and Q from M&S is the only consistent possibility.

Final Assignment Cases:

	F&A	M&S	O&Q	B&H
Case 1	0	P, Q	R	S
Case 2	0	P, Q	S	R

Conclusion:
The confirmed pair of candidates from M&S is Prof. Pakrasi and Prof. Qureshi.

Correct Answer: Option A: Prof. Pakrasi and Prof. Qureshi

Answer 2

Extracted Data and Analysis:

1. Total Votes for Prof. Qureshi: 14 votes

2. Voting Rules:
- Departments vote as a unified bloc.
- Candidates are ineligible to receive votes from their own department.

3. Department Sizes:
- F&A: 9 members
- M&S: 7 members
- O&Q: 5 members
- B&H: 3 members

4. Vote Distribution Analysis:
- Prof. Qureshi’s 14 votes must originate from departments other than his own.
- The largest possible bloc of votes from a single department is 9 (from F&A).
- A plausible scenario for 14 votes is 9 from F&A and 5 from another department.

Conclusion:
- The maximum votes Prof. Qureshi could have received from any single department is 9.

Result: Option 4: (9)

Answer 3

To assess the veracity of statements A and B given that Prof. Samuel is affiliated with the B&H department, let's examine the stipulated conditions:

1. Departmental Constraints:
- Each department has a maximum faculty quota.
- The O&Q department is limited to a single candidate.
- As Prof. Samuel is definitively from B&H, no other individual can be a member of this department.

2. Evaluation of Statement A:
- Statement A posits: "Prof. Pakrasi is from M&S."
- M&S is a substantial department, accommodating one candidate without issue.
- Assigning Prof. Pakrasi to M&S does not contravene any established rules.
⇒ Statement A is substantiated.

3. Evaluation of Statement B:
- Statement B asserts: "Prof. Ramaswamy is from O&Q."
- Given the one-candidate limit for O&Q and Prof. Samuel's B&H assignment, Prof. Ramaswamy can be the sole O&Q representative.
- This arrangement respects all departmental representation requirements.
⇒ Statement B is substantiated.

4. Final Assessment:
- Both statement A and statement B align with the provided conditions.

Consequently, the definitive answer is: Option 2: Both statements A and B.

Answer 4

To determine the candidate from the O&Q department, the following constraints were applied:

1. Department Size and Representation:
- The O&Q department has a maximum of 5 faculty members, indicating it is smaller than the F&A or M&S departments.
- Only one candidate can represent the O&Q department.

2. Elimination Based on Vote Count:
- Prof. Pakrasi and Prof. Qureshi received a significant number of votes, suggesting strong support from larger departments like F&A or M&S.
- Given O&Q's smaller size, it is improbable that either of these professors is affiliated with O&Q.

3. Identifying Potential Candidates:
- The remaining candidates are Prof. Ramaswamy and Prof. Samuel.
- Based on the available information, either of these two individuals is a plausible candidate for the O&Q department.

4. Final Deduction:
- The candidate from the O&Q department is definitively identified as either Prof. Ramaswamy or Prof. Samuel.

Consequently, the correct selection is: Option 1: It was either Prof. Ramaswamy or Prof. Samuel.

Answer 5

Determine the veracity of each statement based on the provided information:

1. Voting Patterns and Departmental Regulations:
- All non-candidates within each department cast votes exclusively for a single candidate originating from a different department.
- Professor Qureshi garnered a considerable number of votes (14), indicating support from multiple departments.

2. Evaluation of Statement A:
- Statement A: "Non-candidates from M&S supported Professor Qureshi."
- If M&S non-candidates voted for Professor Qureshi, this would imply all non-candidates in M&S voted for him (according to the rule).
- However, considering the vote distribution and the necessity for a coherent allocation of total votes, it is improbable that M&S voted for Professor Qureshi.
⇒ Statement A is unlikely to be true.

3. Evaluation of Statement B:
- Statement B: "Non-candidates from F&A voted for Professor Qureshi."
- F&A, being the largest department (9 members), could plausibly contribute a significant portion to Professor Qureshi's 14 votes.
- This scenario is consistent with the premise of Professor Qureshi receiving votes from multiple departments.
⇒ Statement B is likely to be true.

4. Conclusion:
- Only Statement B is substantiated by the data.
Therefore, the correct determination is: Option 4: Only statement B.