`C_0 C_1 +C_1 C_2+C_2 C_3 +...+C_(n-1) C_n`

C_0 C_1 + C_1 C_2 + .... + C_(n-1) C_n = (2^n.n.1.3.5... (2n-1))/(n+1)

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

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + C_(3)x^(3) + …+ C_(n) x^(n) , then C_(0) - (C_(0) - C_(1)) + (C_(0) + C_(1) + C_(2))- (C_(0) + C_(1) + C_(2)+ C_(3)) + ...+ (-1)^(n-1) (C_0) + C_(1) + C_(2) + ...+ C_(n-1)) , when n is even integer is

Show that: C_1/C_0 + 2 C_2/C_1 + 3 C_3/C_2 + .... + n C_n/(C_n -1) = (n(n+1))/2

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

Prove that : C_0 + C_1/2 + C_2/3 + ….. + C_n/(n+1) = (2^(n+1) - 1)/(n+1)

C_0 - [C_1 -2.C_2+ 3.C_3-……..+(-1)^(n-1).n.C_n] =

If (1+x)^n = C_0 + C_1 x+ C_2 x^2 + ….....+ C_n x^n, then C_0+2. C_1 +3. C_2 +….+(n+1) . C_n=

Show that C_0 + (C_0 + C_1) + (C_0 +C_1 + C_2)+……. + (C_0 + C_1 + …+C_n)= (n+2).2^(n-1)