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Math Help - Combinatio Help

  1. #1
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    Combinatio Help

    Suppose that a club consists of 10 men and 13 women. The club is going to form a committee of 8 people. How many 8 person committees have more women than men?

    I thought maybe it was (23!/8!*15!)-(10!/8!*2!) but that does not seem like it gives me the correct answer. Any help would be great!
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  2. #2
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    Hello, ezwind72!

    I don't believe there is a formula for this problem . . .


    Suppose that a club consists of 10 men and 13 women.
    The club is going to form a committee of 8 people.
    How many 8-person committees have more women than men?

    \begin{array}{ccccc}\text{3 men, 5 women} & {10\choose3}{13\choose5} &=& 154,\!440 \\ \\[-3mm]<br />
\text{2 men, 6 women} & {10\choose2}{13\choose6} &=& 77,\!220 \\ \\[-3mm]<br />
\text{1 man, 7 women} & {10\choose1}{13\choose7} &=& 17,\!160 \\ \\[-3mm]<br />
\text{0 men, 8 women} & {10\choose0}{13\choose8} &=& 1,\!287 \\ \\[-3mm] \hline \\[-3mm]<br />
& & \text{Total:} & 250,\!107 \end{array}


    There are \boxed{{\color{blue}250,\!107}} committees with a majority of women.


    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~


    That was such an unusual-looking answer, I just had to run a check.

    \begin{array}{cccc}\text{4 men, 4 women} & {10\choose4}{13\choose4} &=& 150,\!150 \\ \\[-3mm]<br />
\text{5 men, 3 women} & {10\choose5}{13\choose3} &=& 72,\!072 \\ \\[-3mm]<br />
\text{6 men, 2 women} & {10\choose6}{13\choose2} &=& 16,\!380 \\ \\[-3mm]<br />
\text{7 men, 1 woman} & {10\choose7}{13\choose1} &=& 1,\!560 \\ \\[-3mm]<br />
\text{8 men, 0 women} & {10\choose8}{13\choose0} &=& 45 \\ \\[-3mm] \hline \\[-3mm]<br />
& & \text{Total: } & 240,\!207 \end{array}


    Hence, there are: . 250,\!107 + 240,\!207 \:=\:{\color{blue}490,\!314} possible committees.


    Check . There are: . {23\choose8} \;=\;{\color{blue}490,\!314} possible committees . . . YAY!

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  3. #3
    Junior Member
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    Oct 2008
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    I see what you did there. I was wayyyyyyy off! But I now understand how you came about that answer. The formula using summation notation would be (I wish I could say I came up with that solely on my own).



    Thanks for your help!!
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