The number of ways a committee of 8 members can be formed from a group of 10 men and 8 women such that the…
The number of ways a committee of 8 members can be formed from a group of 10 men and 8 women such that the committee contains at most 5 men and at least 5 women is
$8061$
$8612$
$6082$
$8271$
Solution
Since, there are 8 women and 10 men in a group. So, the total number of ways of a committee of 8 member such that committee contains at most 5 men and at least 5 women
$\begin{aligned} & ={ }^8 \mathrm{C}_5{ }^{10} \mathrm{C}_3+{ }^8 \mathrm{C}_6{ }^{10} \mathrm{C}_2+{ }^8 \mathrm{C}_7{ }^{10} \mathrm{C}_1+{ }^8 \mathrm{C}_8{ }^{10} \mathrm{C}_0 \\ & =\frac{8 \times 7 \times 6}{6} \times \frac{10 \times 9 \times 8}{6}+\frac{8 \times 7}{2} \times \frac{10 \times 9}{2}+8 \times 10+1 \times 1 \\ & =6720+1260+81=8061\end{aligned}$