(nC0 + nC1) (nC1 + nC2)(nC2 + nC3)......(nCn-1 + nCn)
= n+1C1 + n+1C2 + n+1C3....n+1Cn
= \(\frac{n+1}1\times\frac{(n+1)n}2\times\frac{(n+1)(n)(n-1)}{3!}....\times\frac{(n+1)}{1!}\)
= (n+1)n\((\frac{n}{n}\times\frac{n(n-1)}{2(n-1)}\times\frac{n(n-1)(n-2)}{3!(n-2)}....\times1)\)
= (n + 1)n x \(\frac{1}{n(n-1)(n-2)....}\times(n\times\frac{n(n-1)}2\times\frac{n(n-1)(n-2)}{3!}\times....\times1)\)
= \(\frac{(n+1)^n}{n!}\)(nC1.nC2.nC3......nCn)
Hence Proved