Prove the following statement by combinatorial argument. Prove the following statement by combinatorial argument:-
$({n \choose0}+{n \choose1})×({n \choose1}+{n \choose2})×\cdots×({n \choose n-1}+{n \choose n})= \frac{{n \choose0}{n \choose1}{n \choose2}\cdots{n \choose n}}{n!}×(1+n)^{n} $

Well I found this question from my friend, a math lover and me too.. i thought that I could do the sum but it was confusing my mind.  I am weak in making combinatorial argument so I need a help to withstand the problem so I posted here.
Hint: 

You can use  $${n\choose k-1}+{n\choose k} = {n+1\choose k}$$

and a fact that $${n+1\choose k}= {(n+1)!\over k!(n+1-k)!} $$ $$={n+1\over n+1-k }\cdot  {n!\over k!(n-k)!} = {n+1\over n+1-k } {n\choose k}$$
Hint: 

You can use  $${n\choose k-1}+{n\choose k} = {n+1\choose k}$$

and a fact that $${n+1\choose k}= {(n+1)!\over k!(n+1-k)!} $$ $$={n+1\over n+1-k }\cdot  {n!\over k!(n-k)!} = {n+1\over n+1-k } {n\choose k}$$