Combinations on a committee of 7, out of group of 25 couples

[jessieputono]

Okay for this question I don't know whether I'm supposed to think of it as 50 people or 25...

A club consists of 25 married couples. Determine how many ways a committee of 7 people can be formed from these couples if
a) there are no restrictions
b) there are exactly 5 men on the committee
c) at least one person must be a man
d) there must be at least 3 women and at least 2 men

Do not simplify answers

 

[stapel]

I would think that you'd need to view this group as having twenty-five women and twenty-five men. That's the only information I think we can use from the fact that this group is formed of couples.

Eliz.

 

[pka]

I will show you the last two without explanation. You should try to find out why these answers work. The apply them to do the first two.

\(\displaystyle \L \begin{array}{l}
\left( {\begin{array}{c}
N \\
k \\
\end{array}} \right) = \frac{{N!}}{{k!\left( {N - k} \right)!}} \\
c)\quad \left( {\begin{array}{c}
{50} \\
7 \\
\end{array}} \right) - \left( {\begin{array}{c}
{25} \\
7 \\
\end{array}} \right) \\
d)\quad \sum\limits_{k = 0}^2 {\left( {\begin{array}{c}
{25} \\
{5 - k} \\
\end{array}} \right)\left( {\begin{array}{c}
{25} \\
{2 + k} \\
\end{array}} \right)} \\
\end{array}\)