Maria and Ariana were only two girls participating in a Billiards tournament. Every Participant played two games with every other participant. The number of games that boys played between themselves proved to exceed by 204, compared to the number of games the boys played with girls. How many participants were there?

Let the number of Boys participating in the tournament be ‘n’ Since, every participant played two games with every other participant, Therefore, the total number of games played among the boys is 2 × nC2 = n(n-1) And the number of games played with each girls = 2n But since there are two girls, hence the total number of games boys played with 2 girls = 2 × 2n = 4n Now, according to the question { n(n-1)} – 4n = 204 n2 - 5n – 204 = 0 n2 - 17n +  12n – 204 = 0 n(n-17)+ 12 (n-17)= 0  n=17,-12  Ignoring negative value We get, n=17  ∴  total number of Participants =  17 + 2 =  19