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  4. There were two women participants in a chess tournament. The number of games the men played between themselves exceeded by 52 the number of games they played with women. If each player played one game with each other, the number of men in the tournament, was

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Q. There were two women participants in a chess tournament. The number of games the men played between themselves exceeded by $52$ the number of games they played with women. If each player played one game with each other, the number of men in the tournament, was

Permutations and Combinations

Solution:

Let there were n men playing in the tournament with $2$ women. According to the given condition,
$ ^{n}C_{2} -\,{}^{n}C_{1} \times\,{}^{2}C_{1} = 52$
$\Rightarrow \frac{n\left(n-1\right)}{2} - 2n = 52$
$\Rightarrow n^{2} - n - 4n = 104 $
$\Rightarrow n^{2} - 5n -104 = 0$
$\Rightarrow n = 13$.