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?

Answer 1

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
$$\begin{gathered} ={}^4{C}_1{\times }^7{C}_4{+}^4{C}_2{\times }^7{C}_3+{}^4{C}_3{\times }^7{C}_2{+}^4{C}_4{\times }^7{C}_1 \\ =140+210+84+7 \\ =441 \end{gathered}$$

(iii) If the team has at least 3 girls, then the number of ways of selecting 5 members
=${}^4{C}_3{\times }^7{C}_2{+}^4{C}_4{\times }^7{C}_1=84+7=91$