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Homework Help: A little help with a binomial theorem proof

  1. Jun 23, 2010 #1
    1. The problem statement, all variables and given/known data
    (here, (n,k) reads n choose k)
    prove that (n,0) - (n, 1) + ... + (-1)n(n,n) = 0

    2. Relevant equations

    binomial theorem

    3. The attempt at a solution
    so this proof is relatively straightforward when n is odd. it's just matching up terms and having them cancel each other out. i'm having a problem proving it when n is even, because each term doesn't match up exactly. and the middle term also alternates between plus or minus depending on whether n/2 is even. (i think i have the middle term is (-1)n(n,n/2).
    but anyway, i've been having trouble with it. a little hint or two would be nice. gracias!
  2. jcsd
  3. Jun 23, 2010 #2
    That's a good approach, but fails for even n, like you noticed.

    Consider directly applying the binomial theorem

    [tex] (x+y)^n = \sum_{i=0}^n (n,i)*x^iy^{n-i}[/tex]

    Now, just pick the right values for [tex] x,y. [/tex]

    Remember the obvious fact that [tex] 1^i = 1,\;\forall i. [/tex]
  4. Jun 24, 2010 #3
    <slaps forehead>
    this is exactly the same as the proof i did before, except for the different x and y values.
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