Question: Committee of 4 people

Comment on Committee of 4 people

In how many ways a committee consisting of 5 men and 6 women can be formed from 8 men and 10 women..
Plz explain me this.
greenlight-admin's picture

Take the task of creating a committee and break it into stages.

STAGE 1: Select the 5 men for the committee
Since the order in which we select the men does not matter, we can use combinations.
We can select 5 men from 8 men in 8C5 ways (56 ways)
So, we can complete stage 1 in 56 ways

STAGE 2: Select the 6 women for the committee
We can select 6 women from 10 women in 10C6 ways (210 ways)
So, we can complete stage 2 in 210 ways

By the Fundamental Counting Principle (FCP), we can complete the 2 stages (and thus create a committee) in (56)(210) ways.

Answer: 11,760 possible committees

In this question, I used the following approach:

1st case: M*4*3*2*1
2nd case: 4*M*3*2*1
3rd case: 4*3*2*M*1
(I have fixed Martha's seat in each case)

And, then I added the product of these cases which is 72. Lastly, I subtract it by 120. What is wrong with this approach?

Thanks in advance Brent!
greenlight-admin's picture

Question link:

There are actually 4 cases in which Martha is NOT in the middle:
1st case: M*4*3*2*1 (= 24 arrangements)
2nd case: 4*M*3*2*1 (= 24 arrangements)
3rd case: 4*3*2*M*1 (= 24 arrangements)
4th case: 4*3*2*1*M (= 24 arrangements)
TOTAL arrangements where Martha is NOT in the middle = 24 + 24 + 24 + 24 = 96

So, TOTAL arrangements where Martha IS in the middle = 120 - 96 = 24

Does that help?


OMG!I can't believe that I made a calculation mistake at the very end. I was getting 14 instead of 24.

Thanks again for all the help and responses!

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