Q20 of 118 Page 18

Let r and n be positive integers such that 1 ≤ r ≤ n. Then prove the following:

nCr + 2nCr – 1 + nCr – 2 = n + 2Cr

Given that we need to prove nCr + 2nCr – 1 + nCr – 2 = n + 2Cr


Consider L.H.S,


We know that nCr + nCr + 1 = n + 1Cr + 1


⇒ nCr + 2nCr – 1 + nCr – 2 = (nCr + nCr – 1) + (nCr – 1 + nCr – 2)


⇒ nCr + 2nCr – 1 + nCr – 2 = n + 1Cr + n + 1Cr – 1


⇒ nCr + 2nCr – 1 + nCr – 2 = n + 2Cr


= R.H.S


∴ L.H.S = R.H.S, thus proved.


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