Question

Two women and some men participated in a chess tournament in which every participant played two games with each of the other participants. If the number of games that the men played between themselves exceeds the number of games that the men played with the women by $66$, then find the number of men who participated in the tournament lies in the interval.

Answer 1

Let there be $x$ number of man and $2$ women.

Since each man play with each other, the combination will be ${}^x{C}_2$

Since there are two games for each men,

$2{\times }^x{C}_2$

No. of ways each men can play with $2$ women $=2x$

Given, $2{\times }^x{C}_2-2x=66⟹x^2-3x-66=0$

Therefore, integer $x$ should be between interval $[10,12)$ to exceed games by $66$.