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A committee of $11$ members is to be formed from $8$ males and $5$ females. if $m$ is the number of ways the commitee is formed with at least $6$ males and $n$ is the number of ways the commitee is formed with at least $3$ females, then :

Option 1)
$m+n=68$

Option 2)
$m=n=78$

Option 3)
$n=m-8$

Option 4)
$m=n=68$

Answers (1)

best_answer

$m=\left ( 6M,5F \right )+\left ( 7M,4F \right )+\left ( 8M,3F \right )$

$=^{8}C_{6}\times^{5} C_{5}+^{8}C_{7}\times^{5} C_{4}+^{8}C_{8}\times ^{5}C_{3}$

$=78$

$m=\left ( 3F,8M \right )+\left ( 4F,7M \right )+\left ( 5F,6M \right )$

$=^{5}C3\times ^{8}C_{8}+^{5}C_{4}\times^{8} C_{7}+^{5}C_{3}\times^{8} C_{3}$

$=78$

$m=n=78$

Option 1)

$m+n=68$

Option 2)

$m=n=78$

Option 3)

$n=m-8$

Option 4)

$m=n=68$

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