Two women and some men participated in a chess tournament in which every participant played two games with…

Let the number of men participating in tournament $=m$.

Number of women participating in tournament $=2$.

Each player will play$2$ games with every other player and two women will play $2$ games with each man.

Total numbers of games played by women $=2\times 2m=4m$

Total number of games played by men $=2{·}^m{C}_2$

From given data

$2{·}^{ m}{C}_2-4m=66$

$\Rightarrow 2\frac{m\left(m-1\right)}{2}-4m=66$

$\Rightarrow m^2-5m-66=0$

$\Rightarrow \left(m-11\right)\left(m+6\right)=0$

$\Rightarrow m=11$ or $-6$ but $m\in N, m\ne -6$

$\Rightarrow m\in \left[10,12\right).$