My Math Forum Committee of 5 members is to be formed

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 April 30th, 2014, 04:44 PM #1 Newbie   Joined: Apr 2014 From: USA Posts: 6 Thanks: 0 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
April 30th, 2014, 05:28 PM   #2
Math Team

Joined: Dec 2006
From: Lexington, MA

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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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