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April 30th, 2014, 05:44 PM   #1
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Committee of 5 members is to be formed

Committee of 5 members to be formed from 5 married couples. Find the # of committees when:

1) the selections random
2) there are more men than women
3) a particular couple is on the committee
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April 30th, 2014, 06:28 PM   #2
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Hello, LuLu99!

Quote:
Committee of 5 members to be formed from 5 married couples.
Find the # of committees when:

(1) the selections are random.

$\displaystyle \quad {10\choose5} \:=\:\frac{10!}{5!\,5!} \:=\:252$ committees.




Quote:
(2) there are more men than women.


$\displaystyle \quad \begin{array}{cccc} \text{3M, 2W} & \text{4M, 1W} & \text{5M, 0W} \\ {5\choose3}{5\choose2} & {5\choose4}{5\choose1} & {5\choose5}{5\choose 0} \\ 10\cdot10 & 5\cdot 5 & 1\cdot 1 \end{array}$

Therefore: $\displaystyle \:100 + 25 + 1 \;=\;126$ committees.




Quote:
(3) a particular couple is on the committee.

If $\displaystyle A$ and $\displaystyle B$ are already on the committee,
we must choose 3 more people from the remaining 8 people.

There are: $\displaystyle \:{8\choose3} \,=\,56$ committees.
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