The total number of games played in chess tournament 
There are two women participating in a chess tournament.
  Every participant played two games with the other participants.
  The number of games that the men played between themselves
  proved to exceed by 66 the number of games that the men 
  played with the women.Find the total number of games played 
  in tournament.

$a.)\ 132\\
b.)\ 112\\
c.)\ 156\\
\color{green}{d.)\ 210} $
Let the number of men be $x$.
$2\dbinom{x}{2}=4x+66 \\
\implies x=11 $
Total games played = $2\dbinom{11+2}{2}=156$.
But the answer given in book is $d.)\ 210 .$
I look for a short and simple way.
I have studied maths up to $12$th grade.

Your translation of the problem is not correct.  Letting $x$ be the number of men and $y$ the number of women, then the men played $2{x \choose 2} = x(x-1)$ games among themselves.  The men played $2xy$ games with the women, so you need $x(x-1)=2xy+66$  If you multiply by $\frac 1x$ you get $x-1=2y+\frac {66}x$, so $x$ must divide $66$.  Try the factors and see what you get.
