A committee of 11 members is to be formed from 8 males and 5 females. If $m$ is the number of ways the…
A committee of 11 members is to be formed from 8 males and 5 females. If $m$ is the number of ways the committee is formed with at least 6 males and $n$ is the number of ways the committee is formed with at least 3 females, then
$\mathrm{m}+\mathrm{n}=68$
$\mathrm{m}=\mathrm{n}=78$
$\quad \mathrm{m}=\mathrm{n}=68$
$\mathrm{n}=\mathrm{m}-8$
Solution
A committee of 11 members is to be formed form 8 males and 5 females.
$\therefore \quad$ When atleast 6 males are included, the committee contains
(6 Males and 5 females), (7 males and 4 females), (8 males and 3 Females)
$\begin{array}{ll}\therefore \quad & \text { Required number of ways } \\ & ={ }^8 \mathrm{C}_6 \times{ }^5 \mathrm{C}_5+{ }^8 \mathrm{C}_7 \times{ }^5 \mathrm{C}_4+{ }^8 \mathrm{C}_8 \times{ }^5 \mathrm{C}_3 \\ \therefore \quad \mathrm{~m}=78 \\ & \text { When atleast } 3 \text { females are included, the } \\ & \text { committee contains } \\ & \text { (3 females and } 8 \text { males), (4 females and } 7 \text { males), } \\ & \text { (5 females and } 6 \text { males) } \\ \therefore \quad & \text { Required number of ways } \\ & ={ }^5 \mathrm{C}_3 \times{ }^8 \mathrm{C}_8+{ }^5 \mathrm{C}_4 \times{ }^8 \mathrm{C}_7+{ }^5 \mathrm{C}_5 \times{ }^8 \mathrm{C}_6 \\ & =78 \\ \therefore \quad & \mathrm{n}=78 \\ \therefore \quad & \mathrm{~m}=\mathrm{n}=78\end{array}$