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  • A committee of 4 people needs to be formed from a group of 8 individuals, including 3 men and 5 women. If the committee must have an equal number of men and women, and tw

A committee of 4 people needs to be formed from a group of 8 individuals, including 3 men and 5 women. If the committee must have an equal number of men and women, and two specific men refuse to serve together, how many different committees can be formed?

 

Option: 1

125


Option: 2

300


Option: 3

100


Option: 4

400


Answers (1)

best_answer

To calculate the number of different committees that can be formed from a group of 8 individuals, including 3 men and 5 women, with the condition that the committee must have an equal number of men and women and two specific men refuse to serve together, we can consider the following:

Since the committee must have an equal number of men and women, we can have either 2 men and 2 women or no men and no women in the committee.

Case 1: 2 men and 2 women in the committee.

To calculate the number of different committees with 2 men and 2 women, we need to select 2 men out of 3 and 2 women out of 5 . The number of ways to do this is \mathrm{ C(3,2) \times C(5,2)=3 \times 10=30.}

Case 2: No men and no women in the committee.

To calculate the number of different committees with no men and no women, we simply need to select 4 individuals out of the remaining 3+5=8 individuals. The number of ways to do this is \mathrm{C(8,4)=70.}

Therefore, the total number of different committees that can be formed is 30+70=100.

Hence, there are 100 different committees that can be formed from the group of 8 individuals, including 3 men and 5 women, with the condition of having an equal number of men and women and two specific men refusing to serve together.

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Divya Prakash Singh

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