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Question

Prove that (ⁿC₀ + ⁿC₁) (ⁿC₁ + ⁿC₂)(ⁿC₂ + ⁿC₃)......(ⁿC_n-1 + ⁿC_n)  = \(\frac{(n+1)^n}{n!}\)(ⁿC₁.ⁿC₂.ⁿC₃......ⁿC_n)

Answer 1

(ⁿC₀ + ⁿC₁) (ⁿC₁ + ⁿC₂)(ⁿC₂ + ⁿC₃)......(ⁿC_n-1 + ⁿC_n)

= ⁿ⁺¹C₁ + ⁿ⁺¹C₂ + ⁿ⁺¹C₃....ⁿ⁺¹C_n

 = \(\frac{n+1}1\times\frac{(n+1)n}2\times\frac{(n+1)(n)(n-1)}{3!}....\times\frac{(n+1)}{1!}\)

 = (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......ⁿC_n)

Hence Proved