C_(0)*C_(3)+C_(1)*C_(4)+C_(2),C_(5)+cdots+C_(n-3)*C_(n)

Find the sum of the following C_(0).C_(3)+C_(1).C_(4)+C_(2).C_(5)+…+C_(n-3).C_(n).

Prove that (C_(0)+C_(1))(C_(1)+C_(2))(C_(2)+C_(3))(C_(n-1)+C_(n))=((n+1)^(n))/(n!)*c_(0)*C_(1)*C_(2).........

If (1+x)^(n)=C_(0)+C_(1)x+C_(2)x^(2)+.....+C_(n)x^(n) then show : C_(0)C_(1)+C_(1)C_(2)+C_(2)C_(3)+.....+C__(n-1)C_(n)=((2n)!)/((n+1)!(n-1)!)

Prove that : For n = 0, 1, 2, 3, ………., n, prove that C_(0).C_(r)+C_(1).C_(r+1)+C_(2).C_(r+2)+….+C_(n-r).C_(n) =""^(2n)C_((n+r)) and hence deduce that C_(0).C_(1)+C_(1).C_(2)+C_(2).C_(3)+……..+C_(n-1).C_(n)=""^(2n)C_(n+1)

(1+x)^(n)=C_(0)+C_(1)x+C_(2)x^(2)+….+C_(n)x^(n) then C_(0)C_(2)+C_(1)C_(3)+C_(2)C_(4)+…..+C_(n-2)C_(n) is equal to :

(1+x)^(n)=C_(0)+C_(1)x+C_(2)x^(2)+….+C_(n)x^(n) then C_(0)C_(2)+C_(1)C_(3)+C_(2)C_(4)+…..+C_(n-2)C_(n) is equal to :

C_(0)C_(2)+C_(1)C_(3)+C_(2)C_(4)+c_(3)C_(5)+...+C_(n-2)C_(n)

(C_(0)+C_(1))(C_(1)+C_(2))(C_(2)+C_(3))(C_(3)+C_(4)).........(C_(n-1)+C_(n))=(C_(0)C_(1)C_(2).....C_(n-1)(n+1)^(n))/(n!)

Prove that C_0.C_3 + C_1.C_4 + C_2.C_5 + …..+C_(n-3).C_n = ""^(2n)C_(n +3)