There are m men and two women participating in a chess tournament....

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,

$2{}^{m}{C}_{2} = 84 + 4m$

$⇒ {m}^{2} - 5m - 84 = 0$

$⇒ {m}^{2} - 12m + 7m - 84 = 0$

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

$⇒ m = 12, - 7$