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Math Help - Combinations and Probability

  1. #1
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    Combinations and Probability

    Question

    A panel of 5 members is drawn from a group consisting of 2 managers, 7 assistants and 4 secretaries.

    (a) How many ways can the panel be formed with at least 1 manager and 2 assistants

    (b) Find the probability that the panel include 2 managers, 2 assistants and 1 secretary



    Answer

    (a) Should I use the following approach?

    1M+2A+2S: C(2,1)*C(7,2)*C(4,2)=252
    1M+3A+1S: C(2,1)*C(7,3)*C(4,1)=280
    1M+4A+0S: C(2,1)*C(7,4)=70
    2M+2A+1S: C(2,2)*C(7,2)*C(4,1)=84
    2M+3A+0S: C(2,2)*C(7,3)=35


    By addition, there are 721 combinations.


    (b) P=2/13*1/12*7/11*6/10*4/9=14/6435
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  2. #2
    Super Member Anonymous1's Avatar
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    Rather than working out all the cases, simply choose the other 2 committee members from the remaining group of 10 people.

    a) {2 \choose 1}{7 \choose 2}{10 \choose 2}

    b) \frac{{2 \choose 2}{7 \choose 2}{4 \choose 1}}{{13\choose 5}}
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  3. #3
    MHF Contributor matheagle's Avatar
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    It's.... at least 1 manager and 2 assistants
    but I'm not sure if that means at least 1 manager and exactly 2 assistants
    or at least 1 manager and at least 2 assistants.

    Prosac's (a) is ok
    In (b) you're putting in order and you need to eliminate that.

    Annoying-mouse's (b) is ok
    You didn't miss the exactly, but I don't think it's correct either.
    Last edited by matheagle; April 5th 2010 at 05:27 AM.
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  4. #4
    Super Member Anonymous1's Avatar
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    Quote Originally Posted by matheagle View Post
    It's.... at least 1 manager and 2 assistants
    but I'm not sure if that means at least 1 manager and exactly 2 assistants
    or at least 1 manager and at least 2 assistants.

    Prosac's (a) is ok
    In (b) you're putting in order and you need to eliminate that.

    Annoying-mouse's (b) is ok
    In (a) you missed the at least, you did exactly 1 ,2, 2.
    Your so dreamy haha.
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  5. #5
    Super Member Anonymous1's Avatar
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    Quote Originally Posted by matheagle View Post
    In (a) you missed the at least, you did exactly 1 ,2, 2.
    Can't the {10\choose 2} be of type manager, assistant or secretary though? And, therein, un-fixing the exactly-ness?
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  6. #6
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    Double confirmed with my teacher and classmates, it should be asking at least 1 manager and at least 2 assistants.

    I got what Ma-the-agleam means. I wrongly put (b) in order. Annoying-mouse's answer is GREAT.

    Thanks to ALL ~

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