The Humanities department of a college is planning to organize eight seminars, one for each of the eight doctoral students - A, B, C, D, E, F, G and H. Four of them are from Economics, three from Sociology and one from Anthropology department. Each student is guided by one among P, Q, R, S and T. Two students are guided by each of P, R and T, while one student is guided by each of Q and S. Each student is guided by a guide belonging to their department.

Each seminar is to be scheduled in one of four consecutive 30-minute slots starting at 9:00 am, 9:30 am, 10:00 am and 10:30 am on the same day. More than one seminars can be scheduled in a slot, provided the guide is free. Only three rooms are available and hence at the most three seminars can be scheduled in a slot. Students who are guided by the same guide must be scheduled in consecutive slots.

The following additional facts are also known.

1. Seminars by students from Economics are scheduled in each of the four slots.

2. A’s is the only seminar that is scheduled at 10:00 am. A is guided by R.

3. F is an Anthropology student whose seminar is scheduled at 10:30 am.

4. The seminar of a Sociology student is scheduled at 9:00 am.

5. B and G are both Sociology students, whose seminars are scheduled in the same slot. The seminar of an Economics student, who is guided by T, is also scheduled in the same slot.

6. P, who is guiding both B and C, has students scheduled in the first two slots.

7. A and G are scheduled in two consecutive slots.

Which one of the following statements is true?

The correct answer is: "Two seminars are scheduled in the first slot."

Let us carefully work through all the given constraints step by step to build the complete schedule and verify this answer.

Step 1 — Establishing the Basic Setup

We have 8 students: A, B, C, D, E, F, G, H.
Departments: 4 from Economics, 3 from Sociology, 1 from Anthropology (F).
Guides: P, Q, R, S, T — where P, R, and T each guide 2 students; Q and S each guide 1 student.
Slots: 9:00 am, 9:30 am, 10:00 am, 10:30 am (Slots 1, 2, 3, 4).
Max 3 seminars per slot (only 3 rooms), and a guide cannot be in two places at once.

Step 2 — Placing Fixed Students from the Clues

From Clue 3: F (Anthropology) is in Slot 4 (10:30 am).

From Clue 2: A's seminar is the only one in Slot 3 (10:00 am). A is guided by R.

From Clue 7: A and G are in two consecutive slots. Since A is in Slot 3, G must be either in Slot 2 or Slot 4.

From Clue 5: B and G (both Sociology) are in the same slot, and an Economics student guided by T is also in that same slot.

From Clue 6: P guides B and C, and P's students are scheduled in the first two slots (Slots 1 and 2). So B is in Slot 1 or Slot 2, and C is in Slot 1 or Slot 2 (in different slots from each other, since guides must be free — a guide cannot attend two seminars at the same time, meaning B and C must be in different consecutive slots).

Step 3 — Narrowing Down G's Slot

From Clue 7, G is in Slot 2 or Slot 4.
From Clue 5, B and G are in the same slot.
From Clue 6, B is in Slot 1 or Slot 2.

The only slot that satisfies both constraints (G in Slot 2 or 4, AND B in Slot 1 or 2, AND B = G's slot) is Slot 2.
Therefore: B and G are both in Slot 2.
And from Clue 6, since B is in Slot 2, C must be in Slot 1 (the other of the first two slots).

Step 4 — Placing the Economics Student Guided by T (from Clue 5)

Clue 5 says the Economics student guided by T is in the same slot as B and G — that is, Slot 2.
Since A is in Slot 3 and is guided by R, and T guides 2 students, the other T-student must also be in a consecutive slot to this T-student (guides' students must be in consecutive slots).
So T's two students are in consecutive slots. One is in Slot 2. The other must be in Slot 1 or Slot 3.

But Slot 3 has only A (the only seminar there, per Clue 2), and A is guided by R, not T. So T's second student must be in Slot 1.

Now, Slot 1 currently has C (guided by P) and a student guided by T (an Economics student).

Step 5 — Placing Sociology and Economics Students Across Slots

From Clue 1: Each of the four slots has at least one Economics student.
From Clue 4: A Sociology student is scheduled in Slot 1 (9:00 am).

Sociology students are B, G, and one more (3 total). B and G are in Slot 2. So the third Sociology student must be in Slot 1 (as required by Clue 4).

The three Sociology students are B, G, and one of {D, E, H}. The one in Slot 1 is from Sociology.

Step 6 — Checking A's Guide (R) and Consecutive Requirement

R guides A (Slot 3) and one other student. R's two students must be in consecutive slots, so R's other student is in Slot 2 or Slot 4.
Slot 2 already has B (guided by P), G, and a T-student — three occupants, which is the maximum. So R's other student is in Slot 4.
This is consistent since Slot 4 has F and now R's other student.

Step 7 — Identifying Guides Q and S

Q and S each guide exactly 1 student.
F (Anthropology, Slot 4) is guided by someone from Anthropology. Only one guide can be the Anthropology guide. Since P guides Sociology students (B and C — but wait, let us verify C's department).

Let's count: P guides B (Sociology) and C. Since "each student is guided by a guide belonging to their department," P must belong to the department of both B and C. B is Sociology, so P is a Sociology guide, meaning C is also Sociology. But we only have 3 Sociology students: B, G, and one other. This means C is the third Sociology student, and the Sociology student in Slot 1 (from Clue 4) is C. This is consistent — C is in Slot 1.

Step 8 — Building the Full Slot Picture

Now let us consolidate:

Slot 1 (9:00 am): C (Sociology, guided by P), + one Economics student guided by T, + the Sociology requirement from Clue 4 is satisfied by C.

Slot 2 (9:30 am): B (Sociology, guided by P), G (Sociology), + one Economics student guided by T. That's 3 students — the slot is full (max 3 rooms).

Slot 3 (10:00 am): A only (Economics, guided by R).

Slot 4 (10:30 am): F (Anthropology), + R's second student (Economics), + possibly others from {D, E, H}.

Remaining students to place: D, E, H (all Economics, since Sociology = B, G, C; Anthropology = F; Economics = A + 3 more from {D, E, H} — all three are Economics).

T's student in Slot 1 is one of D, E, H (Economics). T's student in Slot 2 is one of D, E, H (Economics). That uses 2 of the 3. The remaining one goes to Slot 4.

So Slot 4 has: F + R's second student (one of D/E/H) + possibly one more. But max is 3, and we need each of Slots 1–4 to have at least one Economics student (Clue 1) — Slot 4 already has R's student (Economics), so that's covered.

This places the 3 remaining Economics students (D, E, H) as: one in Slot 1, one in Slot 2, one in Slot 4. This satisfies Clue 1 (Economics in all four slots: Slot 1 ✓, Slot 2 ✓, Slot 3 = A ✓, Slot 4 ✓).

Step 9 — Final Slot Counts

Slot 1: C (Sociology/P) + one of D/E/H (Economics/T) = 2 seminars
Slot 2: B (Sociology/P) + G (Sociology) + one of D/E/H (Economics/T) = 3 seminars
Slot 3: A (Economics/R) = 1 seminar
Slot 4: F (Anthropology) + one of D/E/H (Economics/R) = 2 seminars

Total: 2 + 3 + 1 + 2 = 8 ✓ (all 8 students placed)

Step 10 — Verifying the Answer Against All Options

❌ "Three seminars are scheduled in the last slot" — Slot 4 has 2 seminars. False.
❌ "Only one seminar is scheduled in the second slot" — Slot 2 has 3 seminars. False.
✅ "Two seminars are scheduled in the first slot" — Slot 1 has exactly 2 seminars (C and one Economics/T student). True.
❌ "Three seminars are scheduled in the first slot" �� Slot 1 has only 2 seminars. False.

Conclusion: By systematically applying all the constraints — the consecutive-slot rule for same-guide students, the department-guide alignment, the room limits, and the fixed placements from the numbered clues — we arrive definitively at exactly 2 seminars in Slot 1 (C and one Economics student guided by T). The statement "Two seminars are scheduled in the first slot" is the one that is true.