Choosing teams with minimum number of boys and girls. I need to find different number of teams I can make with 6 people that needs to have at least 2 girls and 2 boys. There are 8 girls and 12 boys. So the way I think is I need to find total number of different teams that I can make first and subtract all boys team, all girls team, one girl 5 boys team, 5 girls one boy team. 
$$\binom{20}{6} - \binom{12}{6} - \binom{8}{6} - \left(\binom{8}{5} \binom{12}{1}\right) -  \left(\binom{8}{1} \binom{12}{5}\right)$$
$$\binom{20}{6} - \binom{12}{6} - \binom{8}{6} - \left( \binom{8}{5} \binom{12}{1}\right) - \left( \binom{8}{1} \binom{12}{5}\right) $$

if boys divided into such as Tom,Jack,Jerry....then
$$X={20\choose 6} - {8\choose6}-{12\choose6}-{8\choose5}{12\choose1}-{12\choose5}{8\choose1}=30800 $$
else it's a naive problem 
just 3 
