Prove that `(C_0+C_1)(C_1+C_2)(C_2+C_3)(C_3+C_4)...........(C_(n-1)+C_n)` = `(C_0C_1C_2.....C_(n-1)(n+1)^n)/(n!)`

If C_(0),C_(1),C_(2)…….,C_(n) are the combinatorial coefficient in the expansion of (1+x)^n, n, ne N , then prove that following C_(1)+2C_(2)+3C_(3)+..+n.C_(n)=n.2^(n-1) C_(0)+2C_(1)+3C_(2)+......+(n+1)C_(n)=(n+2)C_(n)=(n+2)2^(n-1) C_(0),+3C_(1)+5C_(2)+.....+(2n+1)C_n =(n+1)2^n (C_0+C_1)(C_1+C_2)(C_2+C_3)......(C_(n-1)+C_n)=(C_0.C_1.C_2....C_(n-1)(n+1)^n)/(n!) 1.C_0^2+3.C_1^2+....+ (2n+1)C_n^2=((n+1)(2n)!)/(n! n!)

C_1/C_0+2C_2/C_1+3C_3/C_2+............+nC_n/C_(n-1)=(n(n+1))/2

C_0C_2 + C_1C_3 +C_2C_4+……..+C_(n-2) C_n =

If C_(0) , C_(1), C_(2), …, C_(n) are the binomial coefficients in the expansion of (1 + x)^(n) , prove that (C_(0) + 2C_(1) + C_(2) )(C_(1) + 2C_(2) + C_(3))…(C_(n-1) + 2C_(n) + C_(n+1)) ((n-2)^(n))/((n+1)!) prod _(r=1)^(n) (C_(r-1) + C_(r)) .

Prove that C_0C_r+C_1 C_(r+1)+ C_2 C_(r+2)+...............+c_(n-r) C_n=((2n)!)/((n-r)!(n+r)!)

Prove that C_1/C_0+(2c_(2))/C_1+(3C_3)/(C_2)+......+(n.C_n)/(C_(n-1))=(n(n+1))/2

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

Prove that ^n C_0 .^n C_0-^(n+1)C_1 . ^n C_1+^(n+2)C_2 . ^n C_2-=(-1)^ndot

Prove that (C 0 ​ +C 1 ​ )(C 1 ​ +C 2 ​ )(C 2 ​ +C 3 ​ )(C 3 ​ +C 4 ​ ).....(C n−1 ​ +C n ​ )= n! C 0 ​ C 1 ​ C 2 ​ ....C n−1 ​ (n+1) n ​

If (1+x)^(n)=C_(0)+C_(1).x+C_(2).x^(2)+….+C_(n).x^(n). then prove that (i) C_(0)+2C_(1)+3C_(2)+…+(n-1)C_(n)=(n+2).2^(n-1) (ii)C_(0)+3C_(1)+5C_(2)+...+(2n+1)C_(n)=(n+1).2^(n)