Question

Four teams T1, T2, T3, T4 play a tournament where each team plays exactly two matches.
Rules:
1. No match ends in a draw.
2. T1 defeats T3.
3. T4 plays exactly one match before it plays T2.
4. T2 wins exactly one match.
5. T3 does not defeat T4.
6. Total matches = 4.
How many valid sequences of wins/losses across all matches are possible?

• A. 6
• B. 8
• C. 10
• D. 12

Hint:

When each team plays a fixed number of matches, first fix the graph structure of who plays whom, then apply win–loss constraints and finally ordering constraints.

Answer 1

The Correct Option is B

Each team plays exactly two matches, so the 4 matches must pair the teams as follows:
Possible valid match structure:
T1 plays T3 and T4, T2 plays T4 and T3.
This satisfies all degree constraints (each team has degree 2).
Thus the four matches are:
1. T1 vs T3 (T1 wins by rule).
2. T1 vs T4.
3. T2 vs T4.
4. T2 vs T3.
Rule: T4 plays exactly one match before facing T2.
So in the match order, T4 must appear once before the match (T2 vs T4) and once after.
This gives multiple valid orderings of the 4 matches.
Rule: T2 wins exactly one match.
T2's two matches:
• vs T4 → could win or lose.
• vs T3 → could win or lose.
But exactly one must be a win.
Rule: T3 does not defeat T4.
So in match (T3 vs T4), outcome must be:
T4 defeats T3.
Combining all fixed and flexible outcomes:
- T1 defeats T3 (fixed).
- (T4 defeats T3) (required).
- T2 has exactly one win (two possibilities).
- T1 vs T4: free outcome (two possibilities).
Thus:
2 (choices for T1 vs T4) × 2 (choices for which T2 match it wins) = 4 valid result patterns.
But match orderings must also satisfy T4 sequence constraint.
There are exactly 2 valid orderings of the 4 matches that satisfy "T4 plays exactly one match before T2".
Thus total valid sequences = \[ 4 \text{ (valid outcomes)} \times 2 \text{ (valid orders)} = 8. \] Final Answer: 8