A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be selected if the team has
Question:

A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be selected if the team has (i) no girl? (ii) at least one boy and one girl? (iii) at least 3 girls?

Solution:

A group consists of 4 girls and 7 boys. Out of them, 5 are to be selected to form a team.

(i) If the team has no girls, then the number of ways of selecting 5 members $={ }^{7} C_{5}=\frac{7 !}{5 ! 2 !}=\frac{7 \times 6}{2}=21$

(ii) If the team has at least 1 boy and 1 girl, then the number of ways of selecting 5 members

$={ }^{4} C_{1} \times{ }^{7} C_{4}+{ }^{4} C_{2} \times{ }^{7} C_{3}+{ }^{4} C_{3} \times{ }^{7} C_{2}+{ }^{4} C_{4} \times{ }^{7} C_{1}$

$=140+210+84+7$

$=441$

(iii) If the team has at least 3 girls, then the number of ways of selecting 5 members

${ }^{4}{ }^{4} C_{3} \times{ }^{7} C_{2}+{ }^{4} C_{4} \times{ }^{7} C_{1}=84+7=91$