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 ${=}^{12}{C}_0{+}^{12}{C}_1{+}^{12}{C}_2+\dots {+}^{12}{C}_{10}{+}^{12}{C}_{11}{+}^{12}{C}_{12}=2^{12}$
Hence, required number of groups $=2^{12}-\left(\_{}^{12}{C}_0^{}+\_{}^{12}{C}_1^{}+\_{}^{12}{C}_2^{}\right)-2\left({}^6{C}_3{+}^6{C}_4{+}^6{C}_5{+}^6{C}_6\right)$
$=2^{12}-\left({}^{12}{C}_0{+}^{12}{C}_1{+}^{12}{C}_2{+}^6{C}_3\right)$ $-\left({}^6{C}_3{+}^6{C}_4{+}^6{C}_2{+}^6{C}_5{+}^6{C}_1{+}^6{C}_6{+}^6{C}_0\right)$
$=2^{12}-\left(1+12+66+20\right)-2^6=2^{12}-2^7-35$