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.The number of participants & the total numbers of games played in the tournament are respectively

• A. 11,130
• B. 12,144
• C. 13,156
• D. 14,168

Answer 1

Answer: (C)

13,156

let the number of men be $m$.

every participant will play 2 games with other participant.

For a game within men, we have to select any two out of $m$ men.

This can be done in ${}^m{C}_2$ ways.

Since, any pair will play two games between them, total number of games played between men is $2\times {}^m{C}_2=m(m-1)$....(1)

One man can play 2 games with any one woman. So he can play $4$ games with two women. Since, there are $m$ men, number of games played between men and women is $4\times m$....(2)

As per the question,

$m(m-1)=4m+66$

$m^2-m-4m-66=0$

$(m-11)(m+6)=0$

$m=11$ or $m=-6$

m cannot be negative so m is $11$

hence the total number of participants are $11+2=13$ and total number of games 13 players played are $13(13-1)=156$

Hence option C is correct.