There are m men and two women participating in a chess tournament. Each participant plays two games with…

Given, number of men is $m$ and number of women is $2.$

Since each participants competes two games with each other, Hence count of men Vs men total number of game plays $=2{ }^m{C}_2$.

count of men Vs women total number of game plays $=4m$.

As per given condition in question,

$2{ }^m{C}_2=84+4 \mathrm{m}$

$\Rightarrow m^2-5 \mathrm{m}-84=0$

$\Rightarrow m^2-12 \mathrm{m}+7 \mathrm{m}-84=0$

$\Rightarrow \left(m-12\right)\left(m+7\right)=0$.

$\Rightarrow m=12, -7$