30 junior and 38 senior participants participate in a

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