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Math Help - An easy problem!

  1. #1
    MHF Contributor Bruno J.'s Avatar
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    An easy problem!

    Let M(n)=\{-1,\dots, -n\}. Define the empty product to be 1. For every subset of M(n), multiply its elements together and add up the resulting 2^n numbers; what is the result?
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  2. #2
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    Spoiler:
    Given a subset S of M(n), define its twin subset S^* to be S^*=S\cup\{-1\} if -1\notin S and S^*=S\setminus\{-1\} if -1\in S. Then it is clear that the twin subset of S^* is S itself, that all the 2^n subsets of M(n) (provided n\ne0) can be partitioned into 2^{n-1} pairs of mutually twin subsets, and that if the product of all the elements of S is k then the product of all the elements of S^* is -k. Hence the answer to the problem is 0.
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  3. #3
    MHF Contributor Bruno J.'s Avatar
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    Good!

    Here's my solution :

    Spoiler:

    Expand 0=(1-1)(1-2)(1-3)\dots(1-n).
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  4. #4
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    Thanks Bruno.
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