Solution
Understanding the set :-
We are given 4 departments and their faculty members :-
1)Finance and Accounting (F&A) - 9 members
2)Marketing and Strategy (M&S) - 7 members
3)Operations and Quants (O&Q) - 5 members
4)Behaviour and Human Resources (B&H) - 3 members
and 4 candidates, lets call them P, Q, R, S
who were candidates for the post of the Dean of the school.
After reading the set, we can write following conditions.
- There can be either 0 or 1 or 2 candidates form any department.
- A candidate cannot vote for himself or herself.
- Faculty members can not vote for candidate from their own department.
- Non-candidates form same department voted for same candidate.
- P, Q, R, S received 3, 14, 6 and 1 votes.
- There is exactly 1 candidate form OQ.
Firstly we will try to find the number of votes each professor will get :-
| P | Q | R | S | |
|---|---|---|---|---|
| Professor Votes | ||||
| Non-Professor Votes |
Question says, Every faculty member, including the four candidates, voted for the post.
which implies total votes = total of all faculties = 9+7+5+3 = 24
out of which,
4 votes are given by each of P,Q,R,S (let's call them professor votes)-
and other 20 votes by non candidates (let's call them non professor votes).
| P | Q | R | S | |
|---|---|---|---|---|
| Professor Votes | 1 | 1 | 1 | 1 |
| Non-Professor Votes | 2 | 13 | 5 | 0 |
Now, from the set we know that, 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.
therefore we know, that professor votes will be 1 for each of P,Q,R,S
And , P received 3 votes = 1 professor + 2 non professor votes -
Q received 14 votes = 1 professor + 13 non professor votes
R received 6 votes = 1 professor + 5 non professor votes
S received 1 votes = 1 professor + 0 non professor votes
| P | Q | R | S | |
|---|---|---|---|---|
| Professor Votes | 1 | 1 | 1 | 1 |
| Non-Professor Votes | 2 | 13 | 5 | 0 |
| Department 1 | B&H | F&A | M&S | No department |
| Department 2 | 4 or 5 |
We are given in the question that, In each department, all the faculty members who were not candidates voted for the same candidate
therefore, we can say,
P - got two votes are from B&H department
S - he number of non-professor votes is zero, we can infer No department voted for S.
R - the number of non-professor votes is 5 which is from M&S.
Q - have 13 non Professor votes - department of F&A should have voted for him
Two cases are possible here: “9 from F&A and 4 from others” or “8 from F&A and 5 from others”
| P | Q | R | S | |
|---|---|---|---|---|
| Professor Votes | 1 | 1 | 1 | 1 |
| Non-Professor Votes | 2 | 13 | 5 | 0 |
| Department 1 | B&H | F&A | M&S | No department |
| Department 2 | O&H |
Considering case 2;
If number of non-professors votes for Q as “8 from F&A and 5 from other department”
then number of candidates from F&A is 1, B&H is 1, O&Q is 1 and from M&S is 2 which makes total candidates count 5.
but, we know number of candidates is 4
therefore, the case is rejected.
thereore, we have 9 from F&A and 4 from others (which is O&H)
Votes given to each candidate:-
| P | Q | R | S | |
|---|---|---|---|---|
| Professor Votes | 1 | 1 | 1 | 1 |
| Non-Professor Votes | 2 | 13 | 5 | 0 |
| Department 1 | B&H | F&A | M&S | No department |
| Department 2 | O&H |
Candidates and their departments :-
| F&A | |
| M&S | PQ |
| O&Q | R/S |
| B&H | S/R |
Since P voted for R, Q voted for S; R voted for Q and S voted for P the two candidates from the M&S department can be either PQ or RS.
Since Faculty members cannot vote for a candidate from their own department, R cannot be from M&S department since R got votes from M&S department.
Therefore, the two persons from M&S department should be P and Q. R and S can be from either O&Q or B&H.
| P | Q | R | S | |
|---|---|---|---|---|
| Professor Votes | 1 | 1 | 1 | 1 |
| Non-Professor Votes | 2 | 13 | 5 | 0 |
| Department 1 | B&H | F&A | M&S | No department |
| Department 2 | O&H |
| F&A | |
| M&S | PQ |
| O&Q | R/S |
| B&H | S/R |
From the table, it is known that Prof. Pakrasi belongs to M&S. Statement A is true. Since, Prof. Samuel belongs to B&H, Prof. Ramaswamy belongs to O&Q
Hence, the answer is 'Both statements A and B'