A committee of 12 is to be formed from 9 wome | vedclass

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Question

(D) Total committee size is $12$. We have $9$ women and $8$ men.
Condition: At least $5$ women must be included.
Possible cases for (Women,Men) are:

Vedclass · Accepted Answer

$2702, 1008$

Vedclass · Answer

(D) Total committee size is $12$. We have $9$ women and $8$ men.
Condition: At least $5$ women must be included.
Possible cases for (Women,Men) are: $(5, 7), (6, 6), (7, 5), (8, 4), (9, 3)$.
Number of ways = $^9C_5 	imes ^8C_7 + ^9C_6 	imes ^8C_6 + ^9C_7 	imes ^8C_5 + ^9C_8 	imes ^8C_4 + ^9C_9 	imes ^8C_3$
$= (126 	imes 8) + (84 	imes 28) + (36 	imes 56) + (9 	imes 70) + (1 	imes 56)$
$= 1008 + 2352 + 2016 + 630 + 56 = 6062$.
$(i)$ Women are in majority if women count $> 6$ (i.e.,$7, 8, 9$ women):
$= ^9C_7 	imes ^8C_5 + ^9C_8 	imes ^8C_4 + ^9C_9 	imes ^8C_3 = 2016 + 630 + 56 = 2702$.
(ii) Men are in majority if men count $> 6$ (i.e.,$7$ men):
$= ^9C_5 	imes ^8C_7 = 126 	imes 8 = 1008$.

$A$ committee of $12$ is to be formed from $9$ women and $8$ men in which at least $5$ women have to be included in a committee. Then the number of committees in which the women are in majority and men are in majority are respectively

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