Binomial Theorem Proof: (nC0)(mC0) + (nC1)(mC1) + ... + (nCm)(mCm) = (n+m C m)

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Homework Help Overview

The discussion revolves around proving a combinatorial identity involving binomial coefficients, specifically the equation (nC0)(mC0) + (nC1)(mC1) + ... + (nCm)(mCm) = (n+m C m). Participants are exploring methods to establish this identity.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the possibility of using proof by induction as a method to approach the proof. There is a question about whether to perform induction on the variable m and how to structure the inductive step.

Discussion Status

Some guidance has been offered regarding the use of induction, and there seems to be an agreement on the approach to take for the proof. However, the discussion is still in the exploratory phase, with participants clarifying their understanding of the induction process.

Contextual Notes

No specific constraints or missing information have been noted, but the participants are working within the framework of combinatorial proofs and the properties of binomial coefficients.

lifeonfire
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Homework Statement



To Prove:
(nC0)(mC0) + (nC1)(mC1) + ... + (nCm)(mCm) = (n+m C m)

where nC0 = n choose 0 and so on.

Homework Equations





The Attempt at a Solution


Tried expanding the whole thing using factorials - but didn't work. Any hints would be really welcome!
 
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You should review proof by induction and then try to apply it here.
 
Do I do induction on m?? So that would mean , that by assumption ...+(nCm)(mCm) = (n+m C m)...then to prove ...+(nCm+1)(m+1Cm+1) = (n+m+1 C m+1) ...correct?
 
Yes, that should work.
 

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