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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 1 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 1 men must be on the committee, how many different committees can be formed?

 

Option: 1

324

 


Option: 2

246

 


Option: 3

568

 


Option: 4

440


Answers (1)

best_answer

To calculate the number of different committees that can be formed, where at least 1 man must be on the committee, we can consider different scenarios based on the number of men on the committee.

Case 1: Selecting 1 man and 4 women:

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

The number of ways to select 4 women from 6 women is given by \mathrm{C(6,4)=15.}

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

Case 2: 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 * 20=120.}

Case 3: 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 4: 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 * 6=6.}

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

\mathrm{ 60+120+60+6=246 \text {. } }

Therefore, there are 246 different committees that can be formed, where at least 1 man must be on the committee.

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Anam Khan

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