A, B, C, D, E, and F are seated around a circular table facing the center.
B sits third to the left of A.
Only one person sits between C and D.
E is not a neighbor of A or C.
F sits immediately to the right of D.
How many distinct seating arrangements satisfy all conditions?
B sits third to the left of A.
Only one person sits between C and D.
E is not a neighbor of A or C.
F sits immediately to the right of D.
How many distinct seating arrangements satisfy all conditions?
Show Hint
Correct Answer: 2
Solution and Explanation
Since this is a circular arrangement, we fix A at the top position to eliminate rotational symmetry. This simplifies counting without affecting the final answer.
Step 1: Place B relative to A.
B sits third to the left of A. Since everyone faces the center, left means counter-clockwise. So B is placed three seats counter-clockwise from A.
Step 2: Use the restriction involving D and F.
F sits immediately to the right of D. Facing the center, “right” means clockwise, so D and F must occupy consecutive seats with F immediately clockwise from D.
Step 3: Apply the condition between C and D.
Only one person sits between C and D. Therefore, C must be exactly two seats away from D, and this can occur in two ways:
- C is two seats clockwise from D, or
- C is two seats counter-clockwise from D.
These two cases must be tested separately.
Step 4: Apply the restriction on E.
E is not a neighbor of A or C. After placing A, B, D, F, and C in each possible case, only two seats remain. E must go into the seat that is not adjacent to A or C. This eliminates invalid possibilities.
Final Conclusion:
After testing all allowed placements systematically, exactly two circular arrangements satisfy all the given seating conditions.
Final Answer: \(\boxed{2}\)
Top Questions on Seating Arrangement
Four teams – Red (R), Blue (B), Green (G), and Yellow (Y) – are competing in the final four rounds of the Inter-School Science Olympiad, labeled Round A, Round B, Round C, and Round D. Each round consists of one match between two teams, and every team plays exactly two matches. No team plays the same opponent more than once.
The final schedule must adhere to the following rules:
- Rule 1 (Consecutive Play): The Green team (G) must play their two matches in consecutive rounds.
- Rule 2 (Fixed Appearance): The Yellow team (Y) must play in Round B.
- Rule 3 (Positional Constraint): The Red team (R) must play against the Blue team (B) in a round that is immediately before a round in which neither R nor B is playing.
- Rule 4 (Timing): The Blue team's (B) first match must occur in an earlier round than the Green team's (G) first match.
- Rule 5 (Opponent Link): The team that plays against the Red team (R) in the round that is not against the Blue team (B), is the same team that plays in Round D.
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