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  • A committee of 5 people needs to be selected from a group of 10 individuals, including 4 men and 6 women. If at least 2 men must be on the committee, how many different c

A committee of 5 people needs to be selected from a group of 10 individuals, including 4 men and 6 women. If at least 2 men must be on the committee, how many different committees can be formed?

 

Option: 1

525

 


Option: 2

186

 


Option: 3

100

 


Option: 4

360


Answers (1)

best_answer

To calculate the number of different committees that can be formed, we can consider different scenarios based on the number of men on the committee.

Case 1: Selecting 2 men and 3 women:

The number of ways to select 2 men from 4 men is given by \mathrm{C(4,2)=6}

The number of ways to select 3 women from 6 women is given by \mathrm{C(6,3)=20.}

The total number of committees in this case is \mathrm{6 \times 20=120.}

Case 2: Selecting 3 men and 2 women:

The number of ways to select 3 men from 4 men is given by \mathrm{C(4,3)=4.}

The number of ways to select 2 women from 6 women is given by \mathrm{C(6,2)=15.}

The total number of committees in this case is \mathrm{4 \times 15=60.}

Case 3: Selecting 4 men and 1 woman:

The number of ways to select 4 men from 4 men is given by \mathrm{C(4,4)=1.}

The number of ways to select 1 woman from 6 women is given by \mathrm{C(6,1)=6.}

The total number of committees in this case is \mathrm{1 \times 6=6.}

To find the total number of committees that satisfy the given conditions, we sum up the number of committees from each case:

\mathrm{ 120+60+6=186 }

Therefore, there are 186 different committees that can be formed, where at least 2 men must be on the committee.

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