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We know:

nCr + nCr-1 = n+1Cr (1)

n+rCr = nC0 + n+1C1 + n+2C2 + n+3C3 + . . . . . . + n+mCn+1

n+rCr = n+1C0 + n+1C1 + n+2C2 + n+3C3 + . . . . . . + n+mCn+1 (nC0 = n+1C0)

Using equation (1),

n+rCr = n+2C1 + n+2C2 + n+3C3 + . . . . . . + n+mCn+1

n+rCr = n+3C2 + n+3C3 + . . . . . . + n+mCn+1

Proceeding in the same way :

n+rCr = n+mCm-1 + n+mCm = n+m+1Cn+1

n+rCr = n+m+1Cn+1

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