Question

There are $2$ women participating in a chess tournament. Every participant played $2$ games with the other participants. The number of games that the men played between themselves exceeded by $66$ as compared to the number of games that the men played with the women. If the number of participants is $k$ then the value of $k-6$ is

Answer 1

There are 2 women and let number of men are n
Then number of games played by men between themselves is $2\left({}^n{C}_2\right)$
Number of games that the men played with the women is $2\left(2n\right).$
According to question :
$2\left({}^n{C}_2\right)-2\left(2n\right)=66$
$\Rightarrow 2.\frac{n(n-1)}{2}-4n=66$
$\Rightarrow n^2-5n-66=0\Rightarrow (n-11)(n+6)=0$
$\therefore n=11(\because n>0)$
Hence total participants$\left(k\right)=2+11=13$
$\therefore k-6=7$