∑r=0mCn   n+r is equal to:

∑r=0mCn   n+r=∑r=0mCr   n+r ∵n+rCn=n+rCn+r−n=nC0+n+1C1+n+2C2+n+3C3+……+n+mCm Using, nC0=1=n+1C1= n+1C0+n+1C1+n+2C2+n+3C3+….+n+mCm Using this again and again, we are left =Cm   n+m+1=Cn+1   n+m+1