Seven students – P, Q, R, S, T, U and V – had to be allotted project guides from among three professors: Kapoor, Lal and Mehta. Kapoor could guide at most 2 students, Lal at most 2 students and Mehta at most 3 students, so every professor ended up with a full quota.
Each student submitted a strict order of preference over the three professors, shown below (1st choice first).
| Student | 1st | 2nd | 3rd |
|---|---|---|---|
| P | Kapoor | Lal | Mehta |
| Q | Kapoor | Mehta | Lal |
| R | Lal | Kapoor | Mehta |
| S | Kapoor | Mehta | Lal |
| T | Kapoor | Lal | Mehta |
| U | Lal | Kapoor | Mehta |
| V | Mehta | Kapoor | Lal |
The students were processed one at a time in decreasing order of their CGPA (no two students had the same CGPA). When a student's turn came, he or she was allotted the most preferred professor who still had a vacancy.
The following facts are known about the allotment:
1. V had the lowest CGPA.
2. P and U were both allotted their last choice.
3. R was ranked exactly one place below T in the CGPA order.
4. Q had a higher CGPA than S.
5. P had a higher CGPA than U.
Each student submitted a strict order of preference over the three professors, shown below (1st choice first).
| Student | 1st | 2nd | 3rd |
|---|---|---|---|
| P | Kapoor | Lal | Mehta |
| Q | Kapoor | Mehta | Lal |
| R | Lal | Kapoor | Mehta |
| S | Kapoor | Mehta | Lal |
| T | Kapoor | Lal | Mehta |
| U | Lal | Kapoor | Mehta |
| V | Mehta | Kapoor | Lal |
The students were processed one at a time in decreasing order of their CGPA (no two students had the same CGPA). When a student's turn came, he or she was allotted the most preferred professor who still had a vacancy.
The following facts are known about the allotment:
1. V had the lowest CGPA.
2. P and U were both allotted their last choice.
3. R was ranked exactly one place below T in the CGPA order.
4. Q had a higher CGPA than S.
5. P had a higher CGPA than U.
Question 1
Single correct option
Who had the third highest CGPA?
- A. Q
- B. R
- C. S
- D. T
Show answer and solution
Answer: (D) T
Solution. P and U both got Mehta (last choice), so when each of them chose, Kapoor and Lal were already full – i.e. four students ahead of both P and U filled Kapoor and Lal. V is 7th, so P and U are 5th and 6th (P 5th, U 6th by fact 5) and Q, R, S, T occupy ranks 1–4, filling Kapoor (2) and Lal (2). V takes the last Mehta seat (his 1st choice).
Q and S both rank Mehta above Lal; since neither got Mehta, both must have got Kapoor. So Kapoor = {Q, S}, and T (Kapoor > Lal) got Lal, meaning T chose after both Q and S. R got Lal. With R immediately after T: Q, S, T, R in that order (fact 4).
| Rank | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|
| Student | Q | S | T | R | P | U | V |
| Guide | Kapoor | Kapoor | Lal | Lal | Mehta | Mehta | Mehta |
| Choice got | 1st | 1st | 2nd | 1st | 3rd | 3rd | 1st |
Third rank: T.
Q and S both rank Mehta above Lal; since neither got Mehta, both must have got Kapoor. So Kapoor = {Q, S}, and T (Kapoor > Lal) got Lal, meaning T chose after both Q and S. R got Lal. With R immediately after T: Q, S, T, R in that order (fact 4).
| Rank | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|
| Student | Q | S | T | R | P | U | V |
| Guide | Kapoor | Kapoor | Lal | Lal | Mehta | Mehta | Mehta |
| Choice got | 1st | 1st | 2nd | 1st | 3rd | 3rd | 1st |
Third rank: T.
Question 2
Single correct option
Which students were allotted Prof. Lal?
- A. R and U
- B. R and T
- C. T and P
- D. R and Q
Show answer and solution
Answer: (B) R and T
Solution. P and U both got Mehta (last choice), so when each of them chose, Kapoor and Lal were already full – i.e. four students ahead of both P and U filled Kapoor and Lal. V is 7th, so P and U are 5th and 6th (P 5th, U 6th by fact 5) and Q, R, S, T occupy ranks 1–4, filling Kapoor (2) and Lal (2). V takes the last Mehta seat (his 1st choice).
Q and S both rank Mehta above Lal; since neither got Mehta, both must have got Kapoor. So Kapoor = {Q, S}, and T (Kapoor > Lal) got Lal, meaning T chose after both Q and S. R got Lal. With R immediately after T: Q, S, T, R in that order (fact 4).
| Rank | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|
| Student | Q | S | T | R | P | U | V |
| Guide | Kapoor | Kapoor | Lal | Lal | Mehta | Mehta | Mehta |
| Choice got | 1st | 1st | 2nd | 1st | 3rd | 3rd | 1st |
Lal's two students: R and T. U wanted Lal first but both Lal seats were taken before his turn.
Q and S both rank Mehta above Lal; since neither got Mehta, both must have got Kapoor. So Kapoor = {Q, S}, and T (Kapoor > Lal) got Lal, meaning T chose after both Q and S. R got Lal. With R immediately after T: Q, S, T, R in that order (fact 4).
| Rank | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|
| Student | Q | S | T | R | P | U | V |
| Guide | Kapoor | Kapoor | Lal | Lal | Mehta | Mehta | Mehta |
| Choice got | 1st | 1st | 2nd | 1st | 3rd | 3rd | 1st |
Lal's two students: R and T. U wanted Lal first but both Lal seats were taken before his turn.
Question 3
Numerical answer
How many of the seven students were allotted their first choice?
Show answer and solution
Answer: 4
Solution. P and U both got Mehta (last choice), so when each of them chose, Kapoor and Lal were already full – i.e. four students ahead of both P and U filled Kapoor and Lal. V is 7th, so P and U are 5th and 6th (P 5th, U 6th by fact 5) and Q, R, S, T occupy ranks 1–4, filling Kapoor (2) and Lal (2). V takes the last Mehta seat (his 1st choice).
Q and S both rank Mehta above Lal; since neither got Mehta, both must have got Kapoor. So Kapoor = {Q, S}, and T (Kapoor > Lal) got Lal, meaning T chose after both Q and S. R got Lal. With R immediately after T: Q, S, T, R in that order (fact 4).
| Rank | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|
| Student | Q | S | T | R | P | U | V |
| Guide | Kapoor | Kapoor | Lal | Lal | Mehta | Mehta | Mehta |
| Choice got | 1st | 1st | 2nd | 1st | 3rd | 3rd | 1st |
First choices: Q (Kapoor), S (Kapoor), R (Lal), V (Mehta) = 4. T got her 2nd choice; P and U their 3rd.
Q and S both rank Mehta above Lal; since neither got Mehta, both must have got Kapoor. So Kapoor = {Q, S}, and T (Kapoor > Lal) got Lal, meaning T chose after both Q and S. R got Lal. With R immediately after T: Q, S, T, R in that order (fact 4).
| Rank | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|
| Student | Q | S | T | R | P | U | V |
| Guide | Kapoor | Kapoor | Lal | Lal | Mehta | Mehta | Mehta |
| Choice got | 1st | 1st | 2nd | 1st | 3rd | 3rd | 1st |
First choices: Q (Kapoor), S (Kapoor), R (Lal), V (Mehta) = 4. T got her 2nd choice; P and U their 3rd.
Question 4
Numerical answer
Suppose S's preference order had instead been Mehta, Kapoor, Lal, while everything else (all other preferences and the CGPA order) remained the same. How many students would then have been allotted their first choice?
Show answer and solution
Answer: 5
Use the same order Q, S, T, R, P, U, V.
- Q → Kapoor (1st)
- S → Mehta (1st, new preference)
- T → Kapoor (1st; one Kapoor seat is still free) – Kapoor now full
- R → Lal (1st)
- P → Kapoor full, so Lal (2nd) – Lal now full
- U → Lal and Kapoor full, so Mehta (3rd)
- V → Mehta (1st; Mehta has S, U and one seat left)
First choices: Q, S, T, R, V = 5.
Solution. P and U both got Mehta (last choice), so when each of them chose, Kapoor and Lal were already full – i.e. four students ahead of both P and U filled Kapoor and Lal. V is 7th, so P and U are 5th and 6th (P 5th, U 6th by fact 5) and Q, R, S, T occupy ranks 1–4, filling Kapoor (2) and Lal (2). V takes the last Mehta seat (his 1st choice).
Q and S both rank Mehta above Lal; since neither got Mehta, both must have got Kapoor. So Kapoor = {Q, S}, and T (Kapoor > Lal) got Lal, meaning T chose after both Q and S. R got Lal. With R immediately after T: Q, S, T, R in that order (fact 4).
| Rank | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|
| Student | Q | S | T | R | P | U | V |
| Guide | Kapoor | Kapoor | Lal | Lal | Mehta | Mehta | Mehta |
| Choice got | 1st | 1st | 2nd | 1st | 3rd | 3rd | 1st |
- Q → Kapoor (1st)
- S → Mehta (1st, new preference)
- T → Kapoor (1st; one Kapoor seat is still free) – Kapoor now full
- R → Lal (1st)
- P → Kapoor full, so Lal (2nd) – Lal now full
- U → Lal and Kapoor full, so Mehta (3rd)
- V → Mehta (1st; Mehta has S, U and one seat left)
First choices: Q, S, T, R, V = 5.
Solution. P and U both got Mehta (last choice), so when each of them chose, Kapoor and Lal were already full – i.e. four students ahead of both P and U filled Kapoor and Lal. V is 7th, so P and U are 5th and 6th (P 5th, U 6th by fact 5) and Q, R, S, T occupy ranks 1–4, filling Kapoor (2) and Lal (2). V takes the last Mehta seat (his 1st choice).
Q and S both rank Mehta above Lal; since neither got Mehta, both must have got Kapoor. So Kapoor = {Q, S}, and T (Kapoor > Lal) got Lal, meaning T chose after both Q and S. R got Lal. With R immediately after T: Q, S, T, R in that order (fact 4).
| Rank | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|
| Student | Q | S | T | R | P | U | V |
| Guide | Kapoor | Kapoor | Lal | Lal | Mehta | Mehta | Mehta |
| Choice got | 1st | 1st | 2nd | 1st | 3rd | 3rd | 1st |
Question 5
Single correct option
Suppose P had had the highest CGPA of all, with the relative order of the other six students unchanged. Which students would then have been allotted Prof. Mehta?
- A. P, U and V
- B. Q, U and V
- C. S, U and V
- D. S, T and V
Show answer and solution
Answer: (C) S, U and V
New order: P, Q, S, T, R, U, V.
- P → Kapoor (1st)
- Q → Kapoor (1st) – Kapoor full
- S → Kapoor full, so Mehta (2nd)
- T → Lal (2nd)
- R → Lal (1st) – Lal full
- U → Mehta (3rd)
- V → Mehta (1st)
Mehta gets S, U and V.
Solution. P and U both got Mehta (last choice), so when each of them chose, Kapoor and Lal were already full – i.e. four students ahead of both P and U filled Kapoor and Lal. V is 7th, so P and U are 5th and 6th (P 5th, U 6th by fact 5) and Q, R, S, T occupy ranks 1–4, filling Kapoor (2) and Lal (2). V takes the last Mehta seat (his 1st choice).
Q and S both rank Mehta above Lal; since neither got Mehta, both must have got Kapoor. So Kapoor = {Q, S}, and T (Kapoor > Lal) got Lal, meaning T chose after both Q and S. R got Lal. With R immediately after T: Q, S, T, R in that order (fact 4).
| Rank | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|
| Student | Q | S | T | R | P | U | V |
| Guide | Kapoor | Kapoor | Lal | Lal | Mehta | Mehta | Mehta |
| Choice got | 1st | 1st | 2nd | 1st | 3rd | 3rd | 1st |
- P → Kapoor (1st)
- Q → Kapoor (1st) – Kapoor full
- S → Kapoor full, so Mehta (2nd)
- T → Lal (2nd)
- R → Lal (1st) – Lal full
- U → Mehta (3rd)
- V → Mehta (1st)
Mehta gets S, U and V.
Solution. P and U both got Mehta (last choice), so when each of them chose, Kapoor and Lal were already full – i.e. four students ahead of both P and U filled Kapoor and Lal. V is 7th, so P and U are 5th and 6th (P 5th, U 6th by fact 5) and Q, R, S, T occupy ranks 1–4, filling Kapoor (2) and Lal (2). V takes the last Mehta seat (his 1st choice).
Q and S both rank Mehta above Lal; since neither got Mehta, both must have got Kapoor. So Kapoor = {Q, S}, and T (Kapoor > Lal) got Lal, meaning T chose after both Q and S. R got Lal. With R immediately after T: Q, S, T, R in that order (fact 4).
| Rank | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|
| Student | Q | S | T | R | P | U | V |
| Guide | Kapoor | Kapoor | Lal | Lal | Mehta | Mehta | Mehta |
| Choice got | 1st | 1st | 2nd | 1st | 3rd | 3rd | 1st |
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