30 junior and 38 senior participants participate in a tournament. Each pair of juniors plays one match, and each pair of seniors plays one match. Number of boys versus boys matches in juniors is 105, while the number of girls versus girls matches in seniors is 153. If total number of matches played between participants of different genders is 3n + 6n + 6 , then find the value of 'n'.

Solution

Let the number of boys in junior be 'x'. So, ^xC₂ = 105 Or, x(x - 1)/2 = 105 Or, x(x - 1) = 210 Or, x² - x - 210 = 0 Or, x² - 15x + 14x - 210 = 0 Or, x(x - 15) + 14(x - 15) = 0 Or, (x - 15)(x + 14) = 0 So, x = 15 Number of boys in junior = 15 Number of girls in junior = 30 - 15 = 15 Let the number of girls in senior be 'y'. So, ^yC₂ = 153 Or, y(y - 1)/2 = 153 Or, y(y - 1) = 306 Or, y² - y - 306 = 0 Or, y² - 18y + 17y - 306 = 0 Or, y(y - 18) + 17(y - 18) = 0 Or, (y - 18)(y + 17) = 0 So, y = 18 Number of girls in senior = 18 Number of boys in senior = 38 - 18 = 20 Number of matches played between participants of different genders in juniors = 15 × 15 = 225 Number of matches played between participants of different genders in seniors = 20 × 18 = 360 Total number of matches played between participants of different genders = 225 + 360 = 585 According to question, 3n + 6 = 585 Or, 3n = 579 Or, n = 193