KVPY2013MathematicsPermutation and Combination
In a tournament with five teams, each team plays against every other team exctly once. Each game is won by one of the playing teams and the winning team scores one point, while the losing team scores zero. Which of the following is NOT necessarily true?
Options
- AThere are at least two teams which have at most two points each.
- BThere are at least two teams which have at least two points each.
- CThere are at most three teams which have at least threee points each
- DThere are at most four teams which have at most two points each
Correct answer
D. There are at most four teams which have at most two points each
Step-by-step solution
Let teams be T₁, T₂, T₃, T₄ & T₅ Now, we can have 5 teams with the scores of 2 points each matches are (I) T ₁ ~T ₂ (II) T ₁ ~T ₃ (III) T ₁ ~T ₄( IV ) T ₁ ~T ₅ (M) T ₂ ~T ₃( NI ) T ₂ ~T ₄ (MI) T ₂ ~T ₅( VIII ) T ₃ ~T ₄( X ) T ₃ ~T ₅( X ) T ₄ ~T ₅ This score board contradicts, option D D is not always necssarily true.