A committee consisting of atleast three members is to be formed from a group of 6 boys and 6 girls such that it always has a boy and a girl. The number of ways to form such a committee is equal to

Step-by-step solution

Groups having less than three members = 12 C 0 + 12 C 1 + 12 C 2 Groups having only boys = groups having only girls = 6 C 3 + 6 C 4 + 6 C 5 + 6 C 6 Total number of groups = 12 C 0 + 12 C 1 + 12 C 2 + … + 12 C 10 + 12 C 11 + 12 C 12 = 2 12 Hence, required number of groups = 2 12 − C 0 12 + C 1 12 + C 2 12 − 2 6 C 3 + 6 C 4 + 6 C 5 + 6 C 6 = 2 12 − 12 C 0 + 12 C 1 + 12 C 2 + 6 C 3 − 6 C 3 + 6 C 4 + 6 C 2 + 6 C 5 + 6 C 1 + 6 C 6 + 6 C 0 = 2 12 - 1 + 12 + 66 + 20 - 2 6 = 2 12 - 2 7 - 35