Question

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

• A. $2^{12}-2^{7}-13$
• B. $2^{11}-2^{6}-13$
• C. $2^{11}-2^{7}-35$
• D. $2^{12}-2^{7}-35$

Answer 1

Answer: (D)

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 $=^{12}C_{0}+^{12}C_{1}+^{12}C_{2}+\ldots +^{12}C_{10}+^{12}C_{11}+^{12}C_{12}=2^{12}$
Hence, required number of groups $=2^{12}-\left(\_{}^{12}C_{0}^{} + \_{}^{12}C_{1}^{} + \_{}^{12}C_{2}^{}\right)-2\left(^{6} C_{3} +^{6} C_{4} +^{6} C_{5} +^{6} C_{6}\right)$
$=2^{12}-\left(^{12} C_{0} +^{12} C_{1} +^{12} C_{2} +^{6} C_{3}\right)$ $-\left(^{6} C_{3} +^{6} C_{4} +^{6} C_{2} +^{6} C_{5} +^{6} C_{1} +^{6} C_{6} +^{6} C_{0}\right)$
$=2^{12}-\left(1 + 12 + 66 + 20\right)-2^{6}=2^{12}-2^{7}-35$