nCrnCn−r=nrLHS=nCrnCn−r⇒n!(n−r)!r!(n−1)!(n−r)!(r−1)!=nr
Hence proved as LHS=RHS
Let r and n be positive integers such that 1≤r≤n. Then prove the following :
(i) nCrnCr−1=n−r+1r (ii) nn−1Cr−1=(n−r+1)nCr−1 (iii) nCrn−1Cr−1=nr (iv) nCr+2nCr−1+nCr−2=n+2Cr