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