Question

The number of ways in which a committee of $6$ members can be formed from $8$ gentlemen and $4$ ladies so that the committee contains at least $3$ ladies is

• A. $252$
• B. $672$
• C. $444$
• D. $420$

Answer 1

The correct option is A $252$

No. of gentleman $=8$

No. of ladies $=4$

No. of members in committee$=6$ (atleast $3$ ladies)

There are $2$ cases to consider

case $\left(i\right):-3$ ladies in $6$ member committee

No. of ways$=$ (no. of ways of selecting $3$ ladies $\times$

( no. of ways of selecting $3$ gentleman )

${{}^{4}C}_3\times {{}^{8}C}_3\Rightarrow 4\times \frac{8\times 7\times 6}{3\times 2\times 1}\Rightarrow 244$

case $\left(ii\right):-4$ ladies in $6$ member committee

No. of ways$=$ (no. of ways of selecting $4$ ladies)$\times$

(no. of ways of selecting $2$ gentleman)

$={{}^{4}C}_4\times {{}^{8}C}_2=1\times \frac{8\times 7}{2}\Rightarrow 28$

$\therefore$Total no. of ways $=224+28=252$

$\therefore$ the committee can be formed in $252$ different ways