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Summing up binomial coefficients

  1. Mar 28, 2014 #1

    utkarshakash

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    1. The problem statement, all variables and given/known data
    The value of [itex]((^n C_0+^nC_3+........) - \frac{1}{2} (^nC_1+^nC_2+^nC_4+^nC_5+........))^2 + \frac{3}{4} (^nC_1-^nC_2+^nC_4-^nC_5.......)^2
    [/itex]


    3. The attempt at a solution
    I can see that in the left parenthesis, the first bracket contains terms which are multiples of 3 and in the second bracket, those terms are missing. I know it's a much less attempt from my part but that's all I can see. Please help XO
     
  2. jcsd
  3. Mar 28, 2014 #2

    Simon Bridge

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    Sums go to n?
    Have you tried expanding out the squares?
    You'll need extra binomial coefficients.

    The coefficients have a symmetry amongst other properties - you should make a cheat-sheet of the basic properties.

    Also - try n=2 and n=3 and expand them out explicitly and see if you find a pattern.

    BTW: what was the question?
     
  4. Mar 28, 2014 #3

    haruspex

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    Try using powers of a cube root of 1. E.g. look at how the imaginary parts of those change and compare it with the behaviour of the last series.
     
  5. Mar 29, 2014 #4

    utkarshakash

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    I could simplify the first series to
    '[itex]\dfrac{(\omega ^2n + \omega ^n)^2}{4} [/itex]

    But I still can't get the second one. Here's my attempt

    [itex](1+\omega)^n + (1+ \omega ^2)^n = 2(C_0+C_3+.........) - (C_1+C_2+C_4...........)[/itex]
     
  6. Mar 29, 2014 #5

    haruspex

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    Consider other linear combinations of (1+1)n, (1+ω)n, (1+ω2)n.
     
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