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 at least $3$ girls $?$
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 at least $3$ girls $?$
since, the team has to consist of at least $3$ girls, the team can consist of
$(a)$ $3$ girls and $2$ boys, or
$(b)$ $4$ girls and $1$ boy.
Note that the team cannot have all $5$ girls, because, the group has only $4$ girls.
$3$ girls and $2$ boys can be selected in $^{4} C _{3} \times^{7} C _{2}$ ways.
$4$ girls and $1$ boy can be selected in $^{4} C _{4} \times^{7} C _{1}$ ways.
Therefore, the required number of ways
$=\,^{4} C _{3} \times^{7} C _{2}+^{4} C _{4} \times^{7} C _{1}=84+7=91$
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