A committee of 8 has to be formed out of 6 men and 7 women. In how many ways can this be done if the committed
contains.
(i) at least 3 women and at least 3 men?
(ii) Exactly 2 men?
Answers
If the committed contains,
(i) at least 3 women and at least 3 men = 630
(ii) Exactly 2 men = 105
Step-by-step explanation:
Total no. of members in the committee = 8
No. of men = 6
No. of women = 7
Case (i): At least 3 women and at least 3 men in the committee
In order to form a committee having at least 3 women and at least 3 men, the possible combinations are as follows:
3 men & 5 women: ⁶C₃ * ⁷C₅ = [6!/(6-3)!(3)!] * [7!/(7-5)!(5!)] = [5*4] * [7*3] = 20*21 = 420
And
5 men & 3 women: ⁶C₅ * ⁷C₃ = [6!/(6-5)!(5)!] * [7!/(7-3)!(3!)] = [6] * [7*5] = 210
∴ Total no. of ways = 420 + 210 = 630
Case (ii): Exactly 2 men in the committee
Here, in this case, we have to form a committee having exactly 2 men
Therefore,
No. of ways that the committee having exactly 2 men can be formed is,
= ⁶C₂ * ⁷C₆
= [6!/(6-2)!(2)!] * [7!/(7-6)!(6!)]
= [3*5] * [7]
= 15*7
= 105
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