A university admits students to three programmes — Economics, History and Physics — each with exactly 2 seats. Eight applicants, listed below in order of merit rank (Rank 1 is the best), have each listed the programmes they are willing to join, in order of preference. An applicant will never be allotted a programme not on his or her list.
| Rank | Applicant | Preferences (most preferred first) |
|---|---|---|
| 1 | Anu | Physics, Economics |
| 2 | Bala | Economics, Physics, History |
| 3 | Chitra | Physics, History |
| 4 | Dev | Economics, History |
| 5 | Esha | Physics, Economics, History |
| 6 | Farid | History, Economics |
| 7 | Gita | Economics, Physics, History |
| 8 | Hari | Physics, History, Economics |
Allotment is done in rounds. In every round, all seats are allotted afresh: the applicants still in the process are considered one by one in order of rank, and each is allotted the most preferred programme on his or her list that still has a vacant seat in that round (if none, the applicant gets no seat in that round).
- After Round 1, Anu and Dev withdraw from the process.
- After Round 2, Chitra withdraws from the process.
- Round 3 is the final round.
| Rank | Applicant | Preferences (most preferred first) |
|---|---|---|
| 1 | Anu | Physics, Economics |
| 2 | Bala | Economics, Physics, History |
| 3 | Chitra | Physics, History |
| 4 | Dev | Economics, History |
| 5 | Esha | Physics, Economics, History |
| 6 | Farid | History, Economics |
| 7 | Gita | Economics, Physics, History |
| 8 | Hari | Physics, History, Economics |
Allotment is done in rounds. In every round, all seats are allotted afresh: the applicants still in the process are considered one by one in order of rank, and each is allotted the most preferred programme on his or her list that still has a vacant seat in that round (if none, the applicant gets no seat in that round).
- After Round 1, Anu and Dev withdraw from the process.
- After Round 2, Chitra withdraws from the process.
- Round 3 is the final round.
Question 1
Single correct option
Which programme is Esha allotted in the final round?
- A. Physics
- B. Economics
- C. No seat
- D. History
Show answer and solution
Answer: (A) Physics
Round 1: Anu–Physics, Bala–Economics, Chitra–Physics (Physics full), Dev–Economics (Economics full), Esha–History (first two choices full), Farid–History. Gita and Hari get nothing.
Round 2 (Anu, Dev gone): Bala–Economics, Chitra–Physics, Esha–Physics, Farid–History, Gita–Economics, Hari–History (Physics full).
Round 3 (Chitra gone): Bala–Economics, Esha–Physics, Farid–History, Gita–Economics, Hari–Physics.
Esha moves History → Physics → Physics.
Round 2 (Anu, Dev gone): Bala–Economics, Chitra–Physics, Esha–Physics, Farid–History, Gita–Economics, Hari–History (Physics full).
Round 3 (Chitra gone): Bala–Economics, Esha–Physics, Farid–History, Gita–Economics, Hari–Physics.
Esha moves History → Physics → Physics.
Question 2
Numerical answer
How many seats remain vacant after the final round?
Show answer and solution
Answer: 1
Round 3 allotment: Economics — Bala, Gita (full); Physics — Esha, Hari (full); History — Farid only.
Only five applicants remain after the withdrawals of Anu, Dev and Chitra, so with 6 seats at least one must be vacant; the vacancy is in History. Answer: 1.
(Round 2 for reference: Economics — Bala, Gita; Physics — Chitra, Esha; History — Farid, Hari.)
Only five applicants remain after the withdrawals of Anu, Dev and Chitra, so with 6 seats at least one must be vacant; the vacancy is in History. Answer: 1.
(Round 2 for reference: Economics — Bala, Gita; Physics — Chitra, Esha; History — Farid, Hari.)
Question 3
Numerical answer
How many applicants who were allotted a seat in both Round 2 and Round 3 were allotted different programmes in the two rounds?
Show answer and solution
Answer: 1
Round 2 vs Round 3:
| Applicant | Round 2 | Round 3 |
|---|---|---|
| Bala | Economics | Economics |
| Esha | Physics | Physics |
| Farid | History | History |
| Gita | Economics | Economics |
| Hari | History | Physics |
(Chitra had a seat only in Round 2.)
Only Hari changes — Chitra's withdrawal frees a Physics seat, which Hari prefers to History. Answer: 1.
| Applicant | Round 2 | Round 3 |
|---|---|---|
| Bala | Economics | Economics |
| Esha | Physics | Physics |
| Farid | History | History |
| Gita | Economics | Economics |
| Hari | History | Physics |
(Chitra had a seat only in Round 2.)
Only Hari changes — Chitra's withdrawal frees a Physics seat, which Hari prefers to History. Answer: 1.
Question 4
Single correct option
Suppose that, in addition to Anu and Dev, Bala had also withdrawn after Round 1 (everything else unchanged). Which programme would Hari have been allotted in Round 2?
- A. No seat
- B. Economics
- C. History
- D. Physics
Show answer and solution
Answer: (C) History
Round 2 with Anu, Dev and Bala gone (merit order Chitra, Esha, Farid, Gita, Hari):
- Chitra → Physics
- Esha → Physics (Physics now full)
- Farid → History
- Gita → Economics (her first choice; one Economics seat still vacant)
- Hari → Physics full, so History (second seat)
Economics would have one vacant seat, but Hari prefers History to Economics, so he takes History.
- Chitra → Physics
- Esha → Physics (Physics now full)
- Farid → History
- Gita → Economics (her first choice; one Economics seat still vacant)
- Hari → Physics full, so History (second seat)
Economics would have one vacant seat, but Hari prefers History to Economics, so he takes History.
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