Groups having less than three members ${=}^{12}{C}_0{+}^{12}{C}_1{+}^{12}{C}_2$

Groups having only boys $=$ groups having only girls $={\mathrm{}}^6{C}_3+{\mathrm{}}^6{C}_4+{\mathrm{}}^6{C}_5+{\mathrm{}}^6{C}_6$
Total number of groups $={\mathrm{}}^{12}{C}_0+{\mathrm{}}^{12}{C}_1+{\mathrm{}}^{12}{C}_2+\dots +{\mathrm{}}^{12}{C}_{10}+{\mathrm{}}^{12}{C}_{11}+{\mathrm{}}^{12}{C}_{12}=2^{12}$
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)$
$=2^{12}-\left(1+12+66+20\right)-2^6=2^{12}-2^7-35$