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Binomial (Properties of Coefficients)

  1. Aug 16, 2008 #1
    1. The problem statement, all variables and given/known data
    [tex]\sum^{n}_{r=0} (2r+1) (^{n} C_{r})^{2}[/tex]

    3. The attempt at a solution
    If I differentiate this and put x=1;
    I will get the above series without the squares of the binomial coefficients.Will multiplying by [tex](1+x)^{n}[/tex] help now?
  2. jcsd
  3. Aug 17, 2008 #2


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    The "problem" you give is a polynomial. Now, what are you supposed to do with it? What is the question?
  4. Aug 17, 2008 #3
    I have to prove this equal to anyone of these.
    a) (2n+2) ^{2n} C_{n}

    b) (n+1) ^{2n} C_{n}

    c) (2n+1) ^{2n} C_{n}

    d) (n) ^{2n} C_{n}
    Last edited: Aug 17, 2008
  5. Aug 17, 2008 #4
    Help me!!
  6. Aug 18, 2008 #5


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    Well to make things easier (and to cheat a little), let n=2, for example. You should find that only (b) holds. Now of course a proof is required, so that itself doesn't count. But at least you know where to focus your effort.
  7. Aug 19, 2008 #6
    The answer was already given to me in the text book. I am just wondering how to prove the result....
  8. Aug 19, 2008 #7


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    Roughly the same trick as the other one. Replace r in 2r+1 by n-r. As for C(n,r)^2, That's the same as C(n,r)*C(n,n-r). If you sum of over r, isn't that the same as the number of ways to choose n objects from a group of 2n objects?
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