Proving this binomial identity

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    Binomial Identity
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Homework Help Overview

The discussion revolves around proving a binomial identity involving summations and binomial coefficients. The original poster expresses difficulty in deriving one side of the identity from the other, despite understanding a combinatorial proof example.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the nature of the combinatorial proof and question the manipulation of binomial coefficients. There is confusion regarding the reasoning behind the identity and its application to counting specific types of balls.

Discussion Status

The discussion is ongoing, with participants expressing frustration and seeking clarification on the reasoning behind the identity. Some guidance has been offered regarding the combinatorial interpretation, but no consensus has been reached on the proof itself.

Contextual Notes

Participants mention a specific example from a book that illustrates the identity, but there is uncertainty about the manipulation of the binomial terms involved. The original poster indicates a lack of understanding regarding the reasoning presented in the example.

chaotixmonjuish
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\sum_{m=k}^{n-k}\binom{m}{k}\binom{n-m}{k}=\binom{n+1}{2k+1}


I'm not sure how to prove it, I understand the combinatorial proof..i.e. putting it to an example...but i can't derive one side and get the other.
 
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How do you understand the combinatorial proof but not know how to prove it?
 
It was an example given in the book. I just don't see how they manipulated the binomial to move the stuff around.
 
Suppose you had a bag full of n balls. Suppose out of the n balls you had m green ones. Would the right hand side be adding up the ways to count all the gree balls and non-green balls. I didn't even really understand this reasoning. This identity is really frustrating me.
 

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