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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?
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The correct answer is:
There are at most four teams which have at most two points each
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.
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.
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