A committee of 12 persons is to be formed

Asked by admin @ 23/08/2021 in Math viewed by 243 People

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?

Answered by admin @ 23/08/2021

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 !

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