1 + nC1 + n+1C2 + n+2C3+...... + n + r - 1Cr
= nC0 + nC1 + n+1C2 + n+2C3 + ........ + n + r - 1Cr (\(\because\) nC0 = 1)
= n+1C1 + n+1C2 + n+2C3+......n+r-1Cr (\(\because\) nCr + nCr+1 = n+1Cr+1)
= n+2C2 + n+2C3+....n+r-1Cr
= n+r-1Cr-1 + n+r-1Cr
= n+rCr