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Math Help - Combinatorial question

  1. #1
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    Combinatorial question

    Hi.
    I have the following question in this book, and the book's answer to it does not make sense!

    "Think of a group of 20 people. If every one of them shook hands with every other one in the group, how many hand shakes were there?"

    I wrote 19!, since the first person shakes hands with 19 people, the second shakes hands with 18 people (he already shook with the first person, so he has 18 left to shake), the third shakes with 17 and so one...
    Hence 19!
    The book says 190.
    am i missing something here or is it just a typo?
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  2. #2
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    Re: Combinatorial question

    Quote Originally Posted by Stormey View Post
    "Think of a group of 20 people. If every one of them shook hands with every other one in the group, how many hand shakes were there?"
    I wrote 19!, since the first person shakes hands with 19 people, the second shakes hands with 18 people (he already shook with the first person, so he has 18 left to shake), the third shakes with 17 and so one...
    Hence 19! The book says 190.
    It is a combination of 20 taken 2 at a time:
    \binom{20}{2}=\frac{(20)(19)}{(2)(1)}=190.
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  3. #3
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    Re: Combinatorial question

    Ok, thanks Plato.
    I was counting permutations instead of combinations.
    (It should have been 19+18+17+...+1)
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