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?

• A

  There are at least two teams which have at most two points each.
• B

  There are at least two teams which have at least two points each.
• C

  There are at most three teams which have at least threee points each
• Correct answer
  D

  There are at most four teams which have at most two points each

Solution

Let teams be $T_{1}, T_{2}, T_{3}, T_{4} \& T_{5}$ Now, we can have 5 teams with the scores of 2 points each matches are (I) $\mathrm{T}_{1} \mathrm{~T}_{2}$ (II) $\mathrm{T}_{1} \mathrm{~T}_{3}$ (III) $\mathrm{T}_{1} \mathrm{~T}_{4}$ $(\mathrm{IV}) \mathrm{T}_{1} \mathrm{~T}_{5}$ (M) $\mathrm{T}_{2} \mathrm{~T}_{3}$ $(\mathrm{NI}) \mathrm{T}_{2} \mathrm{~T}_{4}$ (MI) $\mathrm{T}_{2} \mathrm{~T}_{5}$ $(\mathrm{VIII}) \mathrm{T}_{3} \mathrm{~T}_{4}$ $(\mathbb{X}) \mathrm{T}_{3} \mathrm{~T}_{5}$ $(\mathrm{X}) \mathrm{T}_{4} \mathrm{~T}_{5}$
This score board contradicts, option D $\therefore D$ is not always necssarily true.

Hence, the correct answer is option: 'D'.