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Binomial (Properties of Coefficients)

  • Thread starter ritwik06
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  • #1
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Homework Statement


[tex]\sum^{n}_{r=0} (2r+1) (^{n} C_{r})^{2}[/tex]


The Attempt at a Solution


[tex]x(1+x^{2})^{n}[/tex]
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?
 

Answers and Replies

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

[tex]
b) (n+1) ^{2n} C_{n}
[/tex]

[tex]
c) (2n+1) ^{2n} C_{n}
[/tex]

[tex]
d) (n) ^{2n} C_{n}
[/tex]
 
Last edited:
  • #4
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Help me!!
 
  • #5
Defennder
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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.
 
  • #6
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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.
The answer was already given to me in the text book. I am just wondering how to prove the result....
 
  • #7
Dick
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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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