A bag contains 7 white, 5 black and 4 red balls. If two balls are drawn at random, find the probability that both the balls are of the same colour.

#### Solution

Out of 16 balls, two balls can be drawn in ^{16}C_{2} ways.

∴ Total number of elementary events = ^{16}C_{2} = 120

'Two balls drawn are of the same colour' means that both are either white or black or red.

Out of seven white balls, two white balls can be drawn in ^{7}C_{2} ways.

Similarly, two black balls can be drawn from five black balls in ^{5}C_{2} ways and two red balls can be drawn from four red balls in ^{4}C_{2} ways.

Therefore, number of ways of drawing two balls of the same colour = ^{7}C_{2} + ^{5}C_{2} + ^{4}C_{2} = 21 + 10 + 6 = 37

i.e. favourable number of ways = 37

Hence, required probability = \[\frac{37}{120}\]