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Simplifaction help?

  1. Jan 17, 2010 #1
    1. The problem statement, all variables and given/known data

    This is actually part of a larger problem that asks us to prove that the number of ways of counting something is equal to [tex]3^n[/tex]. I have worked it out and the equation I get is:

    [tex]\binom{n}{0}2^n + \binom{n}{1}2^{n-1}+\ldots+\binom{n}{n}2^{n-n}[/tex]

    I am wondering how I should simplify this to make it equal to [tex]3^n[/tex]

    2. The attempt at a solution

    I rewrote the above equation into:

    [tex]\displaystyle\sum_{i=0}^{n}\binom{n}{i}2^{n-i}[/tex]

    But then I didn't know how to proceed from here since both the combinatorial choosing term and the powered terms are changing. I also tried factoring out [tex]2^n[/tex] but that didn't do anything.

    Can anyone help me?
    Thanks.
     
  2. jcsd
  3. Jan 17, 2010 #2

    tiny-tim

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    Hi noblerare! :smile:

    Hint: what's (2 + x)n ? :wink:
     
  4. Jan 17, 2010 #3
    Ohhhh, wow. Okay thanks, tiny-tim! Problem solved.
     
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