A committee of 11 members is to be formed
Question:

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:

  1. (1) $m+n=68$

  2. (2) $m=n=78$

  3. (3) $\mathrm{n}=\mathrm{m}-8$

  4. (4) $\mathrm{m}=\mathrm{n}=68$


Correct Option: , 2

Solution:

Since, $m=$ number of ways the committee is formed with at least 6 males

$={ }^{8} \mathrm{C}_{6} \cdot{ }^{5} \mathrm{C}_{5}+{ }^{8} \mathrm{C}_{7} \cdot{ }^{5} \mathrm{C}_{4}+{ }^{8} \mathrm{C}_{8} \cdot{ }^{5} \mathrm{C}_{3}=78$

and $n=$ number of ways the committee is formed with at least 3 females

$={ }^{5} \mathrm{C}_{3} \cdot{ }^{8} \mathrm{C}_{8}+{ }^{5} \mathrm{C}_{4} \cdot{ }^{8} \mathrm{C}_{7}+{ }^{5} \mathrm{C}_{5} \cdot{ }^{8} \mathrm{C}_{6}=78$

Hence $m=n=78$

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