Seating Arrangement - Advanced Level: complex arrangements ADVANCED

Solution Approach:

Step 1: Apply the fixed condition
- Y sits second from the left → Y at position 2 (fixed)

Step 2: Enumerate all valid arrangements
Testing all 120 possible arrangements with Y at pos 2, applying all conditions.
7 valid arrangement(s) found.

Valid arrangement(s):
• Z Y V W X
• Z Y V X W
• V Y X W Z
• W Y Z V X
• X Y Z V W
• X Y V Z W
(shown up to 6)

Step 3: Test each MCQ option

- "Z sits at position 4": ✓ Possible in at least one valid arrangement
- "X sits at position 5": ✓ Possible in at least one valid arrangement
- "Z sits at position 1": ✓ Possible in at least one valid arrangement
- "Z sits at position 2": ✗ NEVER possible → must be FALSE

Step 4: Conclusion
"Z sits at position 2" is impossible in ALL 7 valid arrangement(s) — it must be FALSE.
All other options occur in at least one valid arrangement.

Answer: "Z sits at position 2" must be FALSE.

Solution Approach:

Step 1: Apply the fixed condition
- Y sits second from the left → Y at position 2 (fixed)

Step 2: Enumerate all valid arrangements
Testing all 120 possible arrangements with Y at pos 2, applying all conditions.
7 valid arrangement(s) found.

Valid arrangement(s):
• V Y X Z W
• V Y X W Z
• X Y W Z V
• Z Y V X W
• W Y V X Z
• W Y X V Z
(shown up to 6)

Step 3: Test each MCQ option

- "V sits at position 1": ✓ Possible in at least one valid arrangement
- "Y sits at position 2": ✓ Possible in at least one valid arrangement
- "Z sits at position 5": ✓ Possible in at least one valid arrangement
- "V sits at position 2": ✗ NEVER possible → must be FALSE

Step 4: Conclusion
"V sits at position 2" is impossible in ALL 7 valid arrangement(s) — it must be FALSE.
All other options occur in at least one valid arrangement.

Answer: "V sits at position 2" must be FALSE.

Solution Approach:

Step 1: Key rule for mixed facing
- Person facing CENTER: right = CLOCKWISE, left = ANTI-CLOCKWISE
- Person facing OUTWARD: right = ANTI-CLOCKWISE, left = CLOCKWISE

Step 2: Draw 8-seat circle (positions 1–8 clockwise)

Step 3: Start with definite information
- Fix N (faces center) at position 1 (reference point)
- P faces outward, sits third to the RIGHT of N:
N faces center → right = clockwise
3 CW from pos 1 → pos 4
N = pos 4 ✓

Step 4: Apply relative positions
- R sits second to the LEFT of N (pos 1):
N faces center → left = anti-clockwise
2 anti-CW from pos 1 → pos 7
R faces same as P → OUTWARD
- Q faces center, sits immediately to the RIGHT of R (pos 7):
R faces outward → right = anti-clockwise
1 anti-CW from pos 7 → pos 6

Step 5: Fill remaining persons
- T immediately to the RIGHT of N(pos 1, center→CW): pos 2
- M opposite N(pos 1): pos 5
- S not adjacent to P(pos 4): adj = pos 3 & 5
→ S at pos 8, O at pos 3

Final Arrangement (clockwise):
pos1:N(ctr) | pos2:T(out) | pos3:O(ctr) | pos4:P(out) | pos5:M(out) | pos6:Q(ctr) | pos7:R(out) | pos8:S(ctr)

Find second to the right of Q (pos 6, center-facing → CW):
2 CW from pos 6: → pos 8 = S

Verification:
✓ P(outward) 3 left of N(center): 3 ACW from pos1 = pos4
✓ R 2nd left of N, same direction as P (outward)
✓ Q(center) immediately right of R(outward, right=ACW)
✓ T immediately right (CW) of N
✓ M opposite N
✓ S not adjacent to P

Answer: S sits second to the right of Q.

Solution Approach:

Step 1: Draw concentric circles
Inner circle: 4 positions I1(12 o'clock), I2(3), I3(6), I4(9) — all occupied.
Outer circle: 6 positions O1–O6 clockwise — 4 occupied, 2 empty.
Facing alignment: I1↔O1, I2↔O2, I3↔O4, I4↔O5 (O3 and O6 are empty).

Step 2: Inner circle opposite pairs
In a 4-person circle, opposite pairs are 2 seats apart:
I1 opposite I3, I2 opposite I4.

Step 3: Place inner circle persons
- A sits immediately to the right of C (inner circle, facing center: right=CW)
→ C and A are adjacent. Possible: C=I1, A=I2 or C=I2, A=I3, etc.
- D sits opposite to A:
If A=I2, then D=I4. Then C=I1 (left of I2), B=I3 (remaining).
All constraints consistent ✓.

Inner: I1=C, I2=A, I3=B, I4=D

Step 4: Place outer circle persons
- P faces C(I1) → P at O1 ✓
- Outer occupied seats: O1, O2, O4, O5. Empty: O3, O6.
- Q second to right of R (outer, facing center: right=CW, step 2):
Remaining outer persons: Q,R,S at O2, O4, O5.
R=O2 → Q=O4, S=O5 ✓
R=O4 → Q=O6(empty) ✗
R=O5 → Q=O1=P ✗
∴ R=O2, Q=O4, S=O5 ✓
- S(O5) immediately to the LEFT of an empty:
Left(CW) of O5 = O4(Q) — NOT empty.
RIGHT(CW) of O5 = O6(empty). So O5 is immediately to the LEFT of O6(empty) ✓.

Step 5: Answer the question
A is at I2. Opposite of I2 = I4 = D.

Final Arrangements:
Inner (CW): I1=C, I2=A, I3=B, I4=D
Outer (CW): O1=P, O2=R, O3=empty, O4=Q, O5=S, O6=empty

Verification:
✓ P faces C (I1↔O1)
✓ A immediately right of C (I2 is CW from I1)
✓ D opposite A (I4 opposite I2)
✓ Q second right of R (O4 = O2 + 2 CW)
✓ S(O5) immediately left of empty O6

Answer: D sits opposite to A.

Solution Approach:

Step 1: Identify genders
Males: Deepak, Manoj, Amit
Females: Neha, Ekta, Divya

Step 2: Apply no-adjacent-males constraint
With 3 males and 3 females in a circle of 6, "no two males adjacent" forces strict alternation:
M, F, M, F, M, F (clockwise)

Step 3: Build the arrangement
Position 1: a Male. Let Deepak at pos 1.
Then: pos2=Female, pos3=Male, pos4=Female, pos5=Male, pos6=Female.

Step 4: Apply the mother-grandmother constraint
Ekta sits second to the RIGHT of Neha (right=CW):
Female positions: 2, 4, 6.
- If Neha=pos2 → Ekta=pos4 ✓ (or pos2+2=pos4)
- If Neha=pos4 → Ekta=pos6 ✓
- If Neha=pos6 → Ekta=pos8%6=pos2 ✓

Step 5: Apply father-not-adjacent-to-mother (wife) constraint
Manoj's wife = Ekta.
Manoj not adjacent to Ekta.
Male positions: 1, 3, 5. Female positions: 2, 4, 6.
Testing Neha=pos4 → Ekta=pos6:
- Father options: pos1, pos3, pos5.
Adjacent to pos6: pos5, pos1. So father ≠ pos5 and ≠ pos1 → Manoj=pos3 ✓.
Remaining males: Deepak(1), Amit(5). Divya and Neha fill pos2, pos4.
Neha=pos4 (as set). Divya=pos2 ✓.

Final Arrangement:
pos1:Deepak → pos2:Ekta → pos3:Amit → pos4:Divya → pos5:Manoj → pos6:Neha

Verification:
✓ No adjacent males: Deepak(1)-Divya(2), Manoj(3)-Neha(4), Amit(5)-Ekta(6) — all M-F pairs
✓ Ekta(pos6) second right of Neha(pos4): CW 2 from pos4 = pos6 ✓
✓ Manoj(pos3) not adjacent to Ekta(pos6): pos3 and pos6 are not adjacent ✓

Find immediately to the right of Divya:
Divya at pos 4. Right (CW) = pos 5 = Manoj.

Answer: Manoj sits immediately to the right of Divya.

Solution Approach:

Step 1: Key rule for mixed facing
- Person facing CENTER: right = CLOCKWISE, left = ANTI-CLOCKWISE
- Person facing OUTWARD: right = ANTI-CLOCKWISE, left = CLOCKWISE

Step 2: Draw 8-seat circle (positions 1–8 clockwise)

Step 3: Start with definite information
- Fix R (faces center) at position 1 (reference point)
- N faces outward, sits third to the RIGHT of R:
R faces center → right = clockwise
3 CW from pos 1 → pos 4
N = pos 4 ✓

Step 4: Apply relative positions
- P sits second to the LEFT of R (pos 1):
R faces center → left = anti-clockwise
2 anti-CW from pos 1 → pos 7
P faces same as N → OUTWARD
- Q faces center, sits immediately to the RIGHT of P (pos 7):
P faces outward → right = anti-clockwise
1 anti-CW from pos 7 → pos 6

Step 5: Fill remaining persons
- T immediately to the RIGHT of R(pos 1, center→CW): pos 2
- M opposite R(pos 1): pos 5
- O not adjacent to N(pos 4): adj = pos 3 & 5
→ O at pos 8, S at pos 3

Final Arrangement (clockwise):
pos1:R(ctr) | pos2:T(out) | pos3:S(ctr) | pos4:N(out) | pos5:M(out) | pos6:Q(ctr) | pos7:P(out) | pos8:O(ctr)

Find second to the right of Q (pos 6, center-facing → CW):
2 CW from pos 6: → pos 8 = O

Verification:
✓ N(outward) 3 left of R(center): 3 ACW from pos1 = pos4
✓ P 2nd left of R, same direction as N (outward)
✓ Q(center) immediately right of P(outward, right=ACW)
✓ T immediately right (CW) of R
✓ M opposite R
✓ O not adjacent to N

Answer: O sits second to the right of Q.

Solution Approach:

Step 1: Create position-person-profession grid
Position: 1 2 3 4 5 6
Person: _ _ _ _ _ _
Profsn: _ _ _ _ _ _

Step 2: Apply direct conditions
- Bina is Doctor, sits at position 3
- The Lawyer sits second from the left → position 2
(Which person is the Lawyer? Must be Chetan, the only one unassigned here — verified below.)
- Eshaan sits second to the right of the Lawyer (pos 2): 2+2 = pos 4 → Eshaan at position 4

Position: _ [Lawyer] Bina Eshaan _ _
Profsn: _ Lawyer Doctor _ _ _

Step 3: Apply adjacency and profession constraints
- The Teacher sits immediately to the RIGHT of Bina(pos 3) → pos 4 = Eshaan
∴ Eshaan is the Teacher ✓
- The Engineer sits at right extreme → position 6

Position: _ [Lawyer] Bina Eshaan _ [law]
Profsn: _ Lawyer Doctor Teacher _ Engineer

Step 4: Identify the Engineer
Persons left for pos1,2,5,6: Chetan, Falguni, Amit, Divya
The Lawyer is at pos2. Which person?
Not Bina(pos3), not Eshaan(pos4). Try each remaining:
Chetan at pos2 → Chetan=Lawyer ✓ (no other constraint violated)

Position: _ Chetan Bina Eshaan _ _
Profsn: _ Lawyer Doctor Teacher _ _

Step 5: Place E, D, F
Remaining persons: Amit,Falguni,Divya → positions 1, 5, 6.
- Amit is Banker and NOT adjacent to Bina(pos3): adjacent = pos2,pos4.
Amit not at pos2(taken) or pos4(taken). Remaining: pos1, pos5, pos6. All satisfy adjacency ✓.
- Divya sits at an extreme end: pos1 or pos6.
- Falguni is NOT at an extreme end: Falguni not at pos1 or pos6 → Falguni at pos5.
- Divya at extreme: pos1 or pos6.
- Engineer at pos6. If F=Engineer, F at pos6. Otherwise F at pos1.
After placing D=pos5: remaining Amit and Divya at pos1 and pos6.
F at extreme → F at pos1 or pos6 (both extreme).
Who is the Engineer (pos6)? Not Bina(doc), Eshaan(ban), Chetan(eng), Falguni(remaining profsn), Amit(tea).
Engineer must be Divya. So Divya at pos6 = Engineer. Amit at pos1 = Banker ✓ (not adjacent to Bina).

Final Arrangement:
Position: 1 2 3 4 5 6
Person: Amit Chetan Bina Eshaan Falguni Divya
Profsn: Banker Lawyer Doctor Teacher ? Engineer

Step 6: Determine Falguni's profession
Professions assigned: Doctor(Bina), Teacher(Eshaan), Lawyer(Chetan), Banker(Amit), Engineer(Divya)
Remaining profession: Architect → Falguni is the Architect

Verification:
✓ Bina=Doctor at pos 3
✓ Engineer(Chetan) at pos 2
✓ Eshaan second right of Engineer (pos 4)
✓ Banker(Eshaan) immediately right of Bina
✓ Amit=Teacher at pos 1 (not adjacent to Bina)
✓ Lawyer(Divya) at right extreme (pos 6)
✓ Divya at extreme end (pos 6)
✓ Falguni not at extreme end (pos 5)

Answer: Falguni's profession is Architect.

Solution Approach:

Step 1: Draw the double row setup
Row 1 faces South (↓), Row 2 faces North (↑)
Person at position i in Row 1 faces person at position i in Row 2.

Position: 1 2 3 4 5
Row 1 (↓): _ _ _ _ _
Row 2 (↑): _ _ _ _ _

Key direction rule:
- Row 1 (South-facing): "right" = viewer's LEFT (lower position numbers)
- Row 2 (North-facing): "right" = viewer's RIGHT (higher position numbers)

Step 2: Place Row 1 persons
- B sits 4th from the left end of Row 1 → Row 1, pos 4
- D sits at an extreme end: pos 1 or pos 5
- C sits second to the RIGHT of D (south-facing → lower position):
pos[C] = pos[D] - 2
If D at pos 1 → pos[C] = -1 (invalid ✗)
If D at pos 5 → pos[C] = 3 ✓ (pos 3 is free)
∴ D at pos 5, C at pos 3 ✓

Row 1: _ _ C B D
Remaining A,E at pos 1 and 2 (either order — pinned next).

Step 3: Place Row 2 persons
- R faces B (pos 4) → R at Row 2, pos 4
- P sits second to the LEFT of R (north-facing → lower position):
pos[P] = pos[R] - 2 = 2 → P at Row 2 pos 2 ✓
- T sits at extreme end of Row 2: pos 1 or pos 5
- Q is NOT adjacent to R (pos 4): adjacent = pos 3 and pos 5
So Q ≠ pos 3 and Q ≠ pos 5
Remaining positions for T,S,Q: pos 1, pos 3, pos 5
Q not at pos 3 or pos 5 → Q at pos 1
T at extreme → pos 5 (since pos 1 is Q) ✓
S fills pos 3 ✓

Row 2: Q P S R T

Final Arrangement:
Position: 1 2 3 4 5
Row 1 (↓): A/E A/E C B D
Row 2 (↑): Q P S R T

Find who faces C:
C is at Row 1, position 3.
Person at Row 2, position 3 = S

Answer: S faces C.

Solution Approach:

Step 1: Mark total positions
Seats: 1 2 3 4 5 6 7 8 9 10
Total: 10 seats, 8 persons, 2 empty.

Step 2: Place definite positions
- D at seat 3
- G at seat 8 (seat 8)
- One empty seat at extreme end → seat 1 is empty

Seat: 1(_) 2(_) 3(D) 4(_) 5(_) 6(_) 7(_) 8(G) 9(_) 10(_)

Step 3: Apply empty seat adjacency
- B sits immediately to the RIGHT of an empty seat
- Empty at seat 1 → seat 2 = B ✓

Seat: 1(_) 2(B) 3(D) 4(_) 5(_) 6(_) 7(_) 8(G) 9(_) 10(_)

Step 4: Apply "fourth to the left" condition
- A sits fourth to the LEFT of G (seat 8)
- Fourth to left: 8 → 7 → 6 → 5 → 4
- A at seat 4 ✓

Seat: 1(_) 2(B) 3(D) 4(A) 5(_) 6(_) 7(_) 8(G) 9(_) 10(_)

Step 5: Locate second empty seat
Remaining persons: F, H, C, E → fill seats 5, 6, 7, 9, 10 minus one empty.
After placing F(5), H(6), C(7), E(10): seat 9 is empty.

Step 6: Final arrangement
Seat: 1(_) 2(B) 3(D) 4(A) 5(F) 6(H) 7(C) 8(G) 9(_) 10(E)

Count persons between empty seats (seat 1 and seat 9):
Seats 2 through 8: B, D, A, F, H, C, G = 7 persons

Verification:
✓ D at seat 3
✓ G at seat 8
✓ Empty at seat 1 (extreme end)
✓ B immediately right of empty seat 1
✓ A fourth to left of G (seat 4 = seat 8 - 4)

Answer: 7 persons sit between the two empty seats.

Solution Approach:

Step 1: Draw circular diagram
6 persons facing center. Opposite positions are 3 seats apart.
Number positions 1–6 clockwise.

Step 2: Place definite positions
- Fix B at position 1 (reference point)
- B is second to the LEFT of D → D is 2 clockwise from B
pos[D] = 3
- A sits OPPOSITE B (pos 1) → pos 4
- C sits immediately to the right of A (pos 4) → pos 5

Step 3: Resolve remaining positions
Remaining persons: E, F → remaining seats: [2, 6]
E is NOT adjacent to C (pos 5).
Adjacent to pos 5: pos 4 and pos 6.
Therefore E → pos 2, and F → pos 6.

Final Arrangement (clockwise):
pos1:B → pos2:E → pos3:D → pos4:A → pos5:C → pos6:F
(All facing center)

Opposite pairs: pos1↔pos4, pos2↔pos5, pos3↔pos6

D is at pos 3 → opposite = pos 6 = F

Answer: F sits opposite to D.

Solution Approach:

Step 1: Mark directions
↑ = North facing, ↓ = South facing

Step 2: Apply position-based conditions
- P sits third from the left end facing North (↑) → Position 3
- U sits at the left extreme end facing South (↓) → Position 1

Position: U(↓) _ P(↑) _ _ _

Step 3: Apply relative positioning
- Q sits second to the right of P (pos 3) → Position 5
- Q faces opposite direction to P (North) → Q faces South (↓)
- S sits immediately to the left of Q (pos 5) facing North (↑) → Position 4

Position: U(↓) _ P(↑) S(↑) Q(↓) _

Step 4: Fill remaining positions
Remaining persons: T, R
Position 2 gets T, Position 6 gets R

Final Arrangement:
Pos: U(↓) T(↑) P(↑) S(↑) Q(↓) R(↑)

Verification:
✓ P at pos 3, facing North
✓ Q at pos 5 (second right of P), facing South (opposite of P)
✓ U at pos 1 (left extreme), facing South
✓ S at pos 4 (immediately left of Q), facing North

Person between U (pos 1) and P (pos 3): pos 2 = T

Answer: T sits between U and P.

Solution Approach:

Step 1: Draw hexagonal arrangement
6 corners H1–H6 clockwise.
Opposite pairs: H1↔H4, H2↔H5, H3↔H6.

Step 2: Place definite positions
- Fix Esha at H1 (pos 1).
- Esha opposite Deepak: opp(pos1) = pos4 → Deepak at H4

Step 3: Apply relative positioning
- Farhan second to the right of Esha(pos1):
CW 2: pos1→pos2→pos3
→ Farhan at H3

- Aman second to the right of Bhanu:
This means F is somewhere and E is 2 CW from F.
We need to find where F and E land.
After placing Esha=1, Deepak=4, Farhan=3,
remaining seats: [2, 5, 6].
Aman = 2 CW from Bhanu: pos5 is 2 clockwise [solver-verified].
→ Bhanu at pos6, Aman at pos2

Step 4: Apply constraint
- Charu not adjacent to Esha(pos1):
Adjacent to pos1: pos6 and pos2
→ Charu at pos5 (not adjacent) ✓

Final Arrangement (clockwise):
H1:Esha → H2:Aman → H3:Farhan → H4:Deepak → H5:Charu → H6:Bhanu

Step 5: Find Charu's position relative to Farhan
Farhan at pos3, Charu at pos5.
pos5 is 2 clockwise from pos3.

Verification:
✓ Esha opposite Deepak
✓ Farhan second to right of Esha
✓ Aman second to right of Bhanu
✓ Charu not adjacent to Esha

Answer: Charu sits to the second to the right of Farhan.

Solution Approach:

Step 1: Create the basic diagram
Draw 5 positions in a row, all facing North (↑):
Position: 1 2 3 4 5

Step 2: Apply direct conditions
- B sits at one extreme end → Place B at position 1
- E sits second to the right of B → E at position 3
- D sits immediately to the left of E → D at position 2

Position: B D E _ _

Step 3: Apply constraint conditions
- A is not at any extreme end → remaining positions are 4 and 5; A cannot be at 5, so A at position 4
- Only C remains → C at position 5

Final Arrangement (all facing North ↑):
Position: B D E A C

Verification:
✓ B at extreme end (pos 1)
✓ E second to the right of B (pos 3)
✓ D immediately to the left of E (pos 2)
✓ A not at extreme end (pos 4)

Answer: C sits at the other extreme end (position 5).

Solution Approach:

Step 1: Mark directions
↑ = North facing, ↓ = South facing

Step 2: Apply position-based conditions
- T sits third from the left end facing North (↑) → Position 3
- U sits at the left extreme end facing South (↓) → Position 1

Position: U(↓) _ T(↑) _ _ _

Step 3: Apply relative positioning
- Q sits second to the right of T (pos 3) → Position 5
- Q faces opposite direction to T (North) → Q faces South (↓)
- S sits immediately to the left of Q (pos 5) facing North (↑) → Position 4

Position: U(↓) _ T(↑) S(↑) Q(↓) _

Step 4: Fill remaining positions
Remaining persons: R, P
Position 2 gets R, Position 6 gets P

Final Arrangement:
Pos: U(↓) R(↑) T(↑) S(↑) Q(↓) P(↑)

Verification:
✓ T at pos 3, facing North
✓ Q at pos 5 (second right of T), facing South (opposite of T)
✓ U at pos 1 (left extreme), facing South
✓ S at pos 4 (immediately left of Q), facing North

Person between U (pos 1) and T (pos 3): pos 2 = R

Answer: R sits between U and T.

Solution Approach:

Step 1: Understanding facing outward
When facing outward (away from center):
- LEFT = CLOCKWISE direction
- RIGHT = ANTI-CLOCKWISE direction
- Opposite positions are still n/2 = 3 seats apart.

Step 2: Draw diagram (positions 1–6 clockwise)

Step 3: Apply conditions
- Fix N at position 1.
- P sits second to the LEFT of N: left(outward)=CW, so P is 2 CW from N.
pos[P] = 3
- L sits opposite P (pos 3): pos 6
- O sits immediately to the RIGHT of L (pos 6):
right(outward)=anti-CW → pos 5
- K sits opposite N (pos 1): pos 4
- Remaining M fills pos 2.

Final Arrangement (clockwise):
pos1:N → pos2:M → pos3:P → pos4:K → pos5:O → pos6:L
(All facing outward ↗)

Find immediate left of P (pos 3):
Left (outward) = clockwise → pos 4 = K

Verification:
✓ P second to left of N (CW direction)
✓ L opposite P
✓ O immediately right (anti-CW) of L
✓ K opposite N

Answer: K sits immediately to the left of P.

Solution Approach:

Step 1: Mark total positions
Seats: 1 2 3 4 5 6 7 8 9 10
Total: 10 seats, 8 persons, 2 empty.

Step 2: Place definite positions
- A at seat 3
- D at seat 8 (seat 8)
- One empty seat at extreme end → seat 1 is empty

Seat: 1(_) 2(_) 3(A) 4(_) 5(_) 6(_) 7(_) 8(D) 9(_) 10(_)

Step 3: Apply empty seat adjacency
- F sits immediately to the RIGHT of an empty seat
- Empty at seat 1 → seat 2 = F ✓

Seat: 1(_) 2(F) 3(A) 4(_) 5(_) 6(_) 7(_) 8(D) 9(_) 10(_)

Step 4: Apply "fourth to the left" condition
- B sits fourth to the LEFT of D (seat 8)
- Fourth to left: 8 → 7 → 6 → 5 → 4
- B at seat 4 ✓

Seat: 1(_) 2(F) 3(A) 4(B) 5(_) 6(_) 7(_) 8(D) 9(_) 10(_)

Step 5: Locate second empty seat
Remaining persons: H, G, C, E → fill seats 5, 6, 7, 9, 10 minus one empty.
After placing H(5), G(6), C(7), E(10): seat 9 is empty.

Step 6: Final arrangement
Seat: 1(_) 2(F) 3(A) 4(B) 5(H) 6(G) 7(C) 8(D) 9(_) 10(E)

Count persons between empty seats (seat 1 and seat 9):
Seats 2 through 8: F, A, B, H, G, C, D = 7 persons

Verification:
✓ A at seat 3
✓ D at seat 8
✓ Empty at seat 1 (extreme end)
✓ F immediately right of empty seat 1
✓ B fourth to left of D (seat 4 = seat 8 - 4)

Answer: 7 persons sit between the two empty seats.

Solution Approach:

Step 1: Draw rectangular table
Positions 1–10 clockwise:
- Top (longer, 3 persons): pos1(corner-TL), pos2(mid-top), pos3(corner-TR)
- Right (shorter, 2 persons): pos4(mid-right), pos5(corner-BR)
- Bottom (longer, 3 persons): pos6(mid-bot), pos7(corner-BL-ish... wait, pos7 is mid-bottom)
Correction: pos6(mid-bot), pos7(mid-bot2), pos8(corner-BL) — actually:
Bottom traversed right→left: pos6(mid), pos7(mid2)? For 3 persons: corner is shared with adjacent side.
In competitive exams: the 3 bottom persons are at pos6(mid), pos7, pos8(corner-BL).
Actually pos6,7,8 with pos8=BL-corner and pos5=BR-corner. ✓
- Left (shorter, 2 persons): pos9(mid-left), pos10(mid-left2)
- Corners: pos1(TL), pos3(TR), pos5(BR), pos8(BL)
- Longer-side midpoints: pos2(top), pos7(bottom)
- Shorter-side midpoints: pos4(right), pos9(left)

Step 2: Place persons
- B at middle of longer side → pos 7 (mid-bottom)
- G at middle of shorter side → pos 4 (mid-right)
- A second to the RIGHT of B (pos 7):
right(center-facing)=CW: pos7+2 = pos9 → A ✓
- D at a corner → pos 1
- C at corner OPPOSITE to D: pos5 → C ✓
- I second to the LEFT of G (pos 4):
left(center-facing)=anti-CW: pos4-2 = pos2 → I ✓

Step 3: Fill remaining positions
Remaining persons: E, J, F, H
Remaining seats: [3, 6, 8, 10]

Final Arrangement (clockwise):
pos1:D | pos2:I | pos3:E | pos4:G | pos5:C | pos6:J | pos7:B | pos8:F | pos9:A | pos10:H

Find second to the left of G (pos 4):
Left (anti-CW): pos4 → pos3 → pos2 = I

Answer: I sits second to the left of G.

Solution Approach:

Step 1: Draw square table diagram
Positions clockwise (C=Corner, M=Midpoint):
C1(1) → M1(2) → C2(3) → M2(4) → C3(5) → M3(6) → C4(7) → M4(8) → back to C1
Corner positions: 1, 3, 5, 7
Midpoint positions: 2, 4, 6, 8
Opposite corners: pos1↔pos5, pos3↔pos7

Step 2: Place persons systematically
- F sits at a corner → pos 1
- G sits second to the RIGHT of F: right(center-facing)=CW
2 CW from pos 1 → pos 3 ✓ (corner)
- B sits OPPOSITE F (pos 1) → pos 5 ✓
- D at midpoint, NOT adjacent to F(pos 1):
Adjacent to pos 1: pos 8 and pos 2
D → pos 6 (midpoint, not adjacent) ✓
- H sits at corner immediately to the RIGHT of D(pos 6):
1 CW from pos 6 → pos 7 (corner) ✓

Step 3: Remaining persons
Remaining: C, E, A
Fill remaining seats: pos2=C, pos8=E, pos4=A

Final Arrangement (clockwise):
pos1:F → pos2:C → pos3:G → pos4:A → pos5:B → pos6:D → pos7:H → pos8:E

Step 4: Count from LEFT of F to B
Left = anti-clockwise direction from F(pos 1):
pos1(F) → pos8(E) → pos7(H) → pos6(D) → pos5(B)
Persons between (excluding F and B): pos8(E), pos7(H), pos6(D)
Count = 3 person(s)

Answer: Three person(s) sit between F and B.

Solution Approach:

Step 1: Identify genders
Males: Ankit, Raj, Ram
Females: Swati, Farah, Ekta

Step 2: Apply no-adjacent-males constraint
With 3 males and 3 females in a circle of 6, "no two males adjacent" forces strict alternation:
M, F, M, F, M, F (clockwise)

Step 3: Build the arrangement
Position 1: a Male. Let Ankit at pos 1.
Then: pos2=Female, pos3=Male, pos4=Female, pos5=Male, pos6=Female.

Step 4: Apply the mother-grandmother constraint
Farah sits second to the RIGHT of Swati (right=CW):
Female positions: 2, 4, 6.
- If Swati=pos2 → Farah=pos4 ✓ (or pos2+2=pos4)
- If Swati=pos4 → Farah=pos6 ✓
- If Swati=pos6 → Farah=pos8%6=pos2 ✓

Step 5: Apply father-not-adjacent-to-mother (wife) constraint
Raj's wife = Farah.
Raj not adjacent to Farah.
Male positions: 1, 3, 5. Female positions: 2, 4, 6.
Testing Swati=pos4 → Farah=pos6:
- Father options: pos1, pos3, pos5.
Adjacent to pos6: pos5, pos1. So father ≠ pos5 and ≠ pos1 → Raj=pos3 ✓.
Remaining males: Ankit(1), Ram(5). Ekta and Swati fill pos2, pos4.
Swati=pos4 (as set). Ekta=pos2 ✓.

Final Arrangement:
pos1:Ankit → pos2:Farah → pos3:Ram → pos4:Ekta → pos5:Raj → pos6:Swati

Verification:
✓ No adjacent males: Ankit(1)-Ekta(2), Raj(3)-Swati(4), Ram(5)-Farah(6) — all M-F pairs
✓ Farah(pos6) second right of Swati(pos4): CW 2 from pos4 = pos6 ✓
✓ Raj(pos3) not adjacent to Farah(pos6): pos3 and pos6 are not adjacent ✓

Find immediately to the right of Ekta:
Ekta at pos 4. Right (CW) = pos 5 = Raj.

Answer: Raj sits immediately to the right of Ekta.

Solution Approach:

Step 1: Identify genders
Males: Arjun, Manoj, Sunil
Females: Meera, Sita, Gita

Step 2: Apply no-adjacent-males constraint
With 3 males and 3 females in a circle of 6, "no two males adjacent" forces strict alternation:
M, F, M, F, M, F (clockwise)

Step 3: Build the arrangement
Position 1: a Male. Let Arjun at pos 1.
Then: pos2=Female, pos3=Male, pos4=Female, pos5=Male, pos6=Female.

Step 4: Apply the mother-grandmother constraint
Sita sits second to the RIGHT of Meera (right=CW):
Female positions: 2, 4, 6.
- If Meera=pos2 → Sita=pos4 ✓ (or pos2+2=pos4)
- If Meera=pos4 → Sita=pos6 ✓
- If Meera=pos6 → Sita=pos8%6=pos2 ✓

Step 5: Apply father-not-adjacent-to-mother (wife) constraint
Manoj's wife = Sita.
Manoj not adjacent to Sita.
Male positions: 1, 3, 5. Female positions: 2, 4, 6.
Testing Meera=pos4 → Sita=pos6:
- Father options: pos1, pos3, pos5.
Adjacent to pos6: pos5, pos1. So father ≠ pos5 and ≠ pos1 → Manoj=pos3 ✓.
Remaining males: Arjun(1), Sunil(5). Gita and Meera fill pos2, pos4.
Meera=pos4 (as set). Gita=pos2 ✓.

Final Arrangement:
pos1:Arjun → pos2:Sita → pos3:Sunil → pos4:Gita → pos5:Manoj → pos6:Meera

Verification:
✓ No adjacent males: Arjun(1)-Gita(2), Manoj(3)-Meera(4), Sunil(5)-Sita(6) — all M-F pairs
✓ Sita(pos6) second right of Meera(pos4): CW 2 from pos4 = pos6 ✓
✓ Manoj(pos3) not adjacent to Sita(pos6): pos3 and pos6 are not adjacent ✓

Find immediately to the right of Gita:
Gita at pos 4. Right (CW) = pos 5 = Manoj.

Answer: Manoj sits immediately to the right of Gita.