Question

A team of ten is to be formed from $6$ male doctors and $10$ nurses of whom $5$ are male and $5$ female. In how many ways can this be done, if the team must have at least $4$ doctors and at least $4$ nurses with at least $2$ male nurses and at least $2$ female nurses?

Answer 1

The possible combinations are
(1).$6$ nurses and $4$ doctors.
(2).$5$ nurses and $5$ doctors.
(3.)$4$ nurses and $6$ doctors.
(1) can be further split in to following cases.
3 male and 3 female
2 male and 4 female
4 male and 2 female
${=}^6{C}_4{(}^5{C}_3{\times }^5{C}_3{+}^5{C}_2{\times }^5{C}_4{+}^5{C}_2{\times }^5{C}_4)=15(100+50+50)=3000$
(2) can be further split in to following cases.
2 male and 3 female
3 male and 2 female
${=}^6{C}_5{(}^5{C}_2{\times }^5{C}_3{+}^5{C}_2{\times }^5{C}_3)=6(100+100)=1200$
(3) can be further split in to following cases.
2 male and 2 female
${}^6{C}_5{(}^5{C}_2{\times }^5{C}_2)=100=100$
Adding (1)+(2)+(3)=4300