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If  n Cr, denotes the coefficient of xr in the expansion of (1 + x) n, prove that:

 n Cr + 2(n Cr+1)  + n Cr+2    = n+2  Cr+2

OldTimer  Jan 10, 2018
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Hi Old timer,

This is not a proof, ... I just wanted to convince myself that your statement was true.

 

I considered Pascals triangle which gives the coefficients nCr

Just considering how the pattern works it can be deduced that

 

nCr+nCr+1 n+1Cr+1       and      nCr+1+nCr+2 = n+1Cr+2

 

and

 

n+1Cr+1   +    n+1Cr+2  =    n+2Cr+2

 

so

 

nCr+nCr+1   +    nCr+1+nCr+2   =   n+2Cr+2

 

hence

 

nCr+   2nCr+1  +nCr+2   =   n+2Cr+2

 

 

To prove it I would probably use the fact that    nCr =    n! /(r! * (n-r)! )

But I will admit that I have not worked through this proof.

Melody  Jan 10, 2018

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