A committee of 12 persons is to be formed from 9 women and 8 men. In how many ways can this be done if at least five women have to be included in a committee? In how many of these committees
(a) women are in majority?
(b) men are in majority?
Answers
Answer:
6062
a)2702
b)1008
Step-by-step explanation:
Hi,
Given that there are 9 women and 8 men of which a committee
of 12 persons is to be formed and given that at least 5 women
have to be included.
We can divide the scenario into following cases
Case 1 : 5 W and 7 M , 5 women out of 9 can be selected in ⁹C₅
ways and 7 men out of 8 can be selected in 8 ways, hence total
number of ways this combination of men and women can be
selected are 8*⁹C₅ = 1008
Case 2 : 6 W and 6 M , 6 women out of 9 can be selected in ⁹C₆
ways and 6 men out of 8 can be selected in ⁸C₆ = 28 ways,
hence total number of ways this combination of men and
women can be selected are 28*⁹C₆ = 2352
Case 3 : 7 W and 5 M , 7 women out of 9 can be selected in ⁹C₇
= 36 ways and 5 men out of 8 can be selected in ⁸C₅ = 56 ways,
hence total number of ways this combination of men and
women can be selected are 56*36 = 2016
Case 4 : 8 W and 4 M , 8 women out of 9 can be selected in ⁹C₈
= 9 ways and 4 men out of 8 can be selected in ⁸C₄ = 70 ways,
hence total number of ways this combination of men and
women can be selected are 9*70 = 630
Case 5: 9 W and 3 M , 9 women out of 9 can be selected in 1
way and 3 men out of 8 can be selected in ⁸C₃ = 56 ways, hence
total number of ways this combination of men and women can
be selected are 56 = 56.
Hence, total number of ways would be
(1008 + 2352 + 2016 + 630 + 56)
= 6062.
a) From the above discussion, we can observe that in Case 3 ,
Case 4 and Case 5 women are in majority hence total number of
committees in which women are in majority are
(2016 + 630 + 56) = 2702
b) Only in case 1, men are in majority, hence total number of
committees in which men are in majority are 1008
Hope, it helps !