A jar contains 6 red balls and 'n' green balls. Two balls are drawn at random. If the probability that the two balls drawn are of different colours is 3/7, then find the value of (n − 2)².

ATQ, Total number of balls = 6 + n. Probability that both balls are of different colours = 2 × [6/(n + 6)] × [n/(n + 5)]. ATQ; 2 × [6/(n + 6)] × [n/(n + 5)] = 3/7. Or, 12n/[(n + 6)(n + 5)] = 3/7. Or, 12n × 7 = 3(n + 6)(n + 5). Or, 84n = 3(n² + 11n + 30). Or, 84n = 3n² + 33n + 90. Or, 3n² + 33n + 90 − 84n = 0. Or, 3n² − 51n + 90 = 0. Or, n² − 17n + 30 = 0. Or, n² − 15n − 2n + 30 = 0. Or, n(n − 15) − 2(n − 15) = 0. Or, (n − 15)(n − 2) = 0. So, n = 15 or n = 2. Required value = (n − 2)². For n = 15, (n − 2)² = (15 − 2)² = 13² = 169. For n = 2, (n − 2)² = (2 − 2)² = 0² = 0.