CAT 2023 Slot 1 · DILR · Set 4: Election for the Dean

The full set and all 5 questions (Q16–Q20) from CAT 2023 Slot 1, with answers and solutions.

Faculty members in a management school can belong to one of four departments - Finance and Accounting (F&A), Marketing and Strategy (M&S), Operations and Quants (O&Q) and Behaviour and Human Resources (B&H). The numbers of faculty members in F&A, M&S, O&Q and B&H departments are 9, 7, 5 and 3 respectively.

Prof. Pakrasi, Prof. Qureshi, Prof. Ramaswamy and Prof. Samuel are four members of the school's faculty who were candidates for the post of the Dean of the school. Only one of the candidates was from O&Q.

Every faculty member, including the four candidates, voted for the post. In each department, all the faculty members who were not candidates voted for the same candidate. The rules for the election are listed below.

1. There cannot be more than two candidates from a single department.

2. A candidate cannot vote for himself/herself.

3. Faculty members cannot vote for a candidate from their own department.

After the election, it was observed that Prof. Pakrasi received 3 votes, Prof. Qureshi received 14 votes, Prof. Ramaswamy received 6 votes and Prof. Samuel received 1 vote. Prof. Pakrasi voted for Prof. Ramaswamy, Prof. Qureshi for Prof. Samuel, Prof. Ramaswamy for Prof. Qureshi and Prof. Samuel for Prof. Pakrasi.

Question 16 · Caselet DI

Which two candidates can belong to the same department?
  1. A
    Prof. Pakrasi and Prof. Qureshi
  2. B
    Prof. Pakrasi and Prof. Samuel
  3. C
    Prof. Qureshi and Prof. Ramaswamy
  4. D
    Prof. Ramaswamy and Prof. Samuel
Pick an option, then check.
Answer & solution

Answer: Option A

Solving the set

Step 1 – Votes from candidates: Pakrasi → Ramaswamy, Qureshi → Samuel, Ramaswamy → Qureshi, Samuel → Pakrasi. So each candidate gets exactly one vote from the candidates; the rest come from department blocks (all non-candidates of a department vote together).

Department

Size

Non-candidates if 0 / 1 / 2 candidates

F&A

9

9 / 8 / 7

M&S

7

7 / 6 / 5

O&Q

5

— / 4 / — (exactly one candidate)

B&H

3

3 / 2 / 1

Step 2 – Match the block sizes.

• Samuel: 1 vote, all from Qureshi. No block voted for him.

• Pakrasi: 3 − 1 = 2 block votes. Only a B&H block with one candidate has exactly 2 members. So B&H has one candidate and its block voted for Pakrasi.

• Ramaswamy: 6 − 1 = 5 block votes. A block of 5 is possible only for M&S with two candidates (O&Q has exactly one candidate, so its block is 4).

• Qureshi: 14 − 1 = 13 block votes = F&A (9, no candidate) + O&Q (4).

Candidate

Votes

From candidates

From department blocks

Pakrasi

3

1 (Samuel)

2 = B&H block (B&H has one candidate)

Qureshi

14

1 (Ramaswamy)

13 = F&A (9, no candidate) + O&Q (4)

Ramaswamy

6

1 (Pakrasi)

5 = M&S block (M&S has two candidates)

Samuel

1

1 (Qureshi)

0

Step 3 – Departments. Candidates: two in M&S, one in O&Q, one in B&H, none in F&A. Ramaswamy is not in M&S (M&S votes for him). Rule 3 means a candidate cannot vote for someone from the same department, so Samuel is not with Qureshi or Pakrasi. The M&S pair must therefore be Pakrasi and Qureshi. Ramaswamy and Samuel take O&Q and B&H in some order (Pakrasi is not in B&H, which voted for him; Qureshi is not in O&Q, which voted for him).

Candidate

Department

Votes for

Pakrasi

M&S

Ramaswamy

Qureshi

M&S

Samuel

Ramaswamy

O&Q or B&H

Qureshi

Samuel

B&H or O&Q (the other one)

Pakrasi

This question

From Step 3, Pakrasi and Qureshi are both in M&S. The other pairs are impossible: Ramaswamy and Qureshi (Ramaswamy votes for Qureshi), Pakrasi and Samuel (Samuel votes for Pakrasi), Ramaswamy and Samuel (they are in different departments – O&Q and B&H).

Hence, the answer is option 1.

Question 17 · Caselet DI

Which of the following can be the number of votes that Prof. Qureshi received from a single department?
  1. A
    7
  2. B
    6
  3. C
    8
  4. D
    9
Pick an option, then check.
Answer & solution

Answer: Option D

Solving the set

Step 1 – Votes from candidates: Pakrasi → Ramaswamy, Qureshi → Samuel, Ramaswamy → Qureshi, Samuel → Pakrasi. So each candidate gets exactly one vote from the candidates; the rest come from department blocks (all non-candidates of a department vote together).

Department

Size

Non-candidates if 0 / 1 / 2 candidates

F&A

9

9 / 8 / 7

M&S

7

7 / 6 / 5

O&Q

5

— / 4 / — (exactly one candidate)

B&H

3

3 / 2 / 1

Step 2 – Match the block sizes.

• Samuel: 1 vote, all from Qureshi. No block voted for him.

• Pakrasi: 3 − 1 = 2 block votes. Only a B&H block with one candidate has exactly 2 members. So B&H has one candidate and its block voted for Pakrasi.

• Ramaswamy: 6 − 1 = 5 block votes. A block of 5 is possible only for M&S with two candidates (O&Q has exactly one candidate, so its block is 4).

• Qureshi: 14 − 1 = 13 block votes = F&A (9, no candidate) + O&Q (4).

Candidate

Votes

From candidates

From department blocks

Pakrasi

3

1 (Samuel)

2 = B&H block (B&H has one candidate)

Qureshi

14

1 (Ramaswamy)

13 = F&A (9, no candidate) + O&Q (4)

Ramaswamy

6

1 (Pakrasi)

5 = M&S block (M&S has two candidates)

Samuel

1

1 (Qureshi)

0

Step 3 – Departments. Candidates: two in M&S, one in O&Q, one in B&H, none in F&A. Ramaswamy is not in M&S (M&S votes for him). Rule 3 means a candidate cannot vote for someone from the same department, so Samuel is not with Qureshi or Pakrasi. The M&S pair must therefore be Pakrasi and Qureshi. Ramaswamy and Samuel take O&Q and B&H in some order (Pakrasi is not in B&H, which voted for him; Qureshi is not in O&Q, which voted for him).

Candidate

Department

Votes for

Pakrasi

M&S

Ramaswamy

Qureshi

M&S

Samuel

Ramaswamy

O&Q or B&H

Qureshi

Samuel

B&H or O&Q (the other one)

Pakrasi

This question

Qureshi's 14 votes: 9 from F&A, 4 from O&Q and 1 from Ramaswamy. The only option that matches a single department's votes is 9 (F&A).

Hence, the answer is option 4.

Question 18 · Caselet DI

If Prof. Samuel belongs to B&H, which of the following statements is/are true?

Statement A: Prof. Pakrasi belongs to M&S.

Statement B: Prof. Ramaswamy belongs to O&Q

  1. A
    Neither statement A nor statement B
  2. B
    Only statement B
  3. C
    Only statement A
  4. D
    Both statements A and B
Pick an option, then check.
Answer & solution

Answer: Option D

Solving the set

Step 1 – Votes from candidates: Pakrasi → Ramaswamy, Qureshi → Samuel, Ramaswamy → Qureshi, Samuel → Pakrasi. So each candidate gets exactly one vote from the candidates; the rest come from department blocks (all non-candidates of a department vote together).

Department

Size

Non-candidates if 0 / 1 / 2 candidates

F&A

9

9 / 8 / 7

M&S

7

7 / 6 / 5

O&Q

5

— / 4 / — (exactly one candidate)

B&H

3

3 / 2 / 1

Step 2 – Match the block sizes.

• Samuel: 1 vote, all from Qureshi. No block voted for him.

• Pakrasi: 3 − 1 = 2 block votes. Only a B&H block with one candidate has exactly 2 members. So B&H has one candidate and its block voted for Pakrasi.

• Ramaswamy: 6 − 1 = 5 block votes. A block of 5 is possible only for M&S with two candidates (O&Q has exactly one candidate, so its block is 4).

• Qureshi: 14 − 1 = 13 block votes = F&A (9, no candidate) + O&Q (4).

Candidate

Votes

From candidates

From department blocks

Pakrasi

3

1 (Samuel)

2 = B&H block (B&H has one candidate)

Qureshi

14

1 (Ramaswamy)

13 = F&A (9, no candidate) + O&Q (4)

Ramaswamy

6

1 (Pakrasi)

5 = M&S block (M&S has two candidates)

Samuel

1

1 (Qureshi)

0

Step 3 – Departments. Candidates: two in M&S, one in O&Q, one in B&H, none in F&A. Ramaswamy is not in M&S (M&S votes for him). Rule 3 means a candidate cannot vote for someone from the same department, so Samuel is not with Qureshi or Pakrasi. The M&S pair must therefore be Pakrasi and Qureshi. Ramaswamy and Samuel take O&Q and B&H in some order (Pakrasi is not in B&H, which voted for him; Qureshi is not in O&Q, which voted for him).

Candidate

Department

Votes for

Pakrasi

M&S

Ramaswamy

Qureshi

M&S

Samuel

Ramaswamy

O&Q or B&H

Qureshi

Samuel

B&H or O&Q (the other one)

Pakrasi

This question

If Samuel is in B&H, Ramaswamy must be in O&Q (Step 3). Pakrasi is always in M&S. Both statements are true.

Hence, the answer is option 4.

Question 19 · Caselet DI

What best can be concluded about the candidate from O&Q?
  1. A
    It was Prof. Samuel.
  2. B
    It was either Prof. Ramaswamy or Prof. Samuel.
  3. C
    It was Prof. Ramaswamy.
  4. D
    It was either Prof. Pakrasi or Prof. Qureshi.
Pick an option, then check.
Answer & solution

Answer: Option B

Solving the set

Step 1 – Votes from candidates: Pakrasi → Ramaswamy, Qureshi → Samuel, Ramaswamy → Qureshi, Samuel → Pakrasi. So each candidate gets exactly one vote from the candidates; the rest come from department blocks (all non-candidates of a department vote together).

Department

Size

Non-candidates if 0 / 1 / 2 candidates

F&A

9

9 / 8 / 7

M&S

7

7 / 6 / 5

O&Q

5

— / 4 / — (exactly one candidate)

B&H

3

3 / 2 / 1

Step 2 – Match the block sizes.

• Samuel: 1 vote, all from Qureshi. No block voted for him.

• Pakrasi: 3 − 1 = 2 block votes. Only a B&H block with one candidate has exactly 2 members. So B&H has one candidate and its block voted for Pakrasi.

• Ramaswamy: 6 − 1 = 5 block votes. A block of 5 is possible only for M&S with two candidates (O&Q has exactly one candidate, so its block is 4).

• Qureshi: 14 − 1 = 13 block votes = F&A (9, no candidate) + O&Q (4).

Candidate

Votes

From candidates

From department blocks

Pakrasi

3

1 (Samuel)

2 = B&H block (B&H has one candidate)

Qureshi

14

1 (Ramaswamy)

13 = F&A (9, no candidate) + O&Q (4)

Ramaswamy

6

1 (Pakrasi)

5 = M&S block (M&S has two candidates)

Samuel

1

1 (Qureshi)

0

Step 3 – Departments. Candidates: two in M&S, one in O&Q, one in B&H, none in F&A. Ramaswamy is not in M&S (M&S votes for him). Rule 3 means a candidate cannot vote for someone from the same department, so Samuel is not with Qureshi or Pakrasi. The M&S pair must therefore be Pakrasi and Qureshi. Ramaswamy and Samuel take O&Q and B&H in some order (Pakrasi is not in B&H, which voted for him; Qureshi is not in O&Q, which voted for him).

Candidate

Department

Votes for

Pakrasi

M&S

Ramaswamy

Qureshi

M&S

Samuel

Ramaswamy

O&Q or B&H

Qureshi

Samuel

B&H or O&Q (the other one)

Pakrasi

This question

From Step 3, Ramaswamy and Samuel share O&Q and B&H, and nothing decides which is where. So the O&Q candidate is either Ramaswamy or Samuel.

Hence, the answer is option 2.

Question 20 · Caselet DI

Which of the following statements is/are true?

Statement A: Non-candidates from M&S voted for Prof. Qureshi.

Statement B: Non-candidates from F&A voted for Prof. Qureshi.

  1. A
    Both statements A and B
  2. B
    Only statement B
  3. C
    Only statement A
  4. D
    Neither statement A nor statement B
Pick an option, then check.
Answer & solution

Answer: Option B

Solving the set

Step 1 – Votes from candidates: Pakrasi → Ramaswamy, Qureshi → Samuel, Ramaswamy → Qureshi, Samuel → Pakrasi. So each candidate gets exactly one vote from the candidates; the rest come from department blocks (all non-candidates of a department vote together).

Department

Size

Non-candidates if 0 / 1 / 2 candidates

F&A

9

9 / 8 / 7

M&S

7

7 / 6 / 5

O&Q

5

— / 4 / — (exactly one candidate)

B&H

3

3 / 2 / 1

Step 2 – Match the block sizes.

• Samuel: 1 vote, all from Qureshi. No block voted for him.

• Pakrasi: 3 − 1 = 2 block votes. Only a B&H block with one candidate has exactly 2 members. So B&H has one candidate and its block voted for Pakrasi.

• Ramaswamy: 6 − 1 = 5 block votes. A block of 5 is possible only for M&S with two candidates (O&Q has exactly one candidate, so its block is 4).

• Qureshi: 14 − 1 = 13 block votes = F&A (9, no candidate) + O&Q (4).

Candidate

Votes

From candidates

From department blocks

Pakrasi

3

1 (Samuel)

2 = B&H block (B&H has one candidate)

Qureshi

14

1 (Ramaswamy)

13 = F&A (9, no candidate) + O&Q (4)

Ramaswamy

6

1 (Pakrasi)

5 = M&S block (M&S has two candidates)

Samuel

1

1 (Qureshi)

0

Step 3 – Departments. Candidates: two in M&S, one in O&Q, one in B&H, none in F&A. Ramaswamy is not in M&S (M&S votes for him). Rule 3 means a candidate cannot vote for someone from the same department, so Samuel is not with Qureshi or Pakrasi. The M&S pair must therefore be Pakrasi and Qureshi. Ramaswamy and Samuel take O&Q and B&H in some order (Pakrasi is not in B&H, which voted for him; Qureshi is not in O&Q, which voted for him).

Candidate

Department

Votes for

Pakrasi

M&S

Ramaswamy

Qureshi

M&S

Samuel

Ramaswamy

O&Q or B&H

Qureshi

Samuel

B&H or O&Q (the other one)

Pakrasi

This question

From Step 2, the M&S block (5) voted for Ramaswamy, so statement A is false. The F&A block (9) voted for Qureshi, so statement B is true.

Hence, the answer is option 2.

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