Prove that `^nC_0-^nC_1+^nC_2-^nC_3+....+(-1)^r ^C_r+....=(-1)^(r-1) ^(n-1)C_(r-1)`

Prove that ""^nC_0+2*""^nC_1+3*""^nC_2+...+(n+1)""^nC_n=(n+2)2^(n-1)

Prove that ""^nC_0-2*""^nC_1+3*""^nC_2-...+(-1)""^n(n+1)""^nC_n=0

Prove that ""^nC_0+3*""^nC_1+5*""^nC_2+...+(2n+1)""^nC_n=(n+1)2^(n)

The value of ""^0C_0-^nC_1+^nC_2-....+ -1^(n^n)C_n is

Prove that "^nC_r + 2. ^nC_(r-1) + ^nC_(r-2) = ^(n+2)C_r

If n is a positive integer then (1+x)^n=^nC_0 x^0+^nC_1 x^1+^nC_2^2+………+^nC_rx^r= sum _(r=0)^n ^nC_rx^r and (1-x)^n= ^nC_0x^0-^nC_1 x^1 +^nC_2-^nC_3x^3+………+(-1)^n ^nC_nx^n=sum_(r=0)^n (-1)^r ^nC_r x^r On the basis of above information answer the following question:If n is a positive integer then lim_nrarroo n[^nc_n- 2/3 . ^nC_(n-1)+(2/3)^2.^nC_(n-2-...........+(-1)^n(2/3)^n.^nC_n]= (A) 1 (B) 1/2 (C) 0 (D) 1/3

If n is a positive integer then (1+x)^n=^nC_0 x^0+^nC_1 x^1+^nC_2^2+………+^nC_rx^r= sum _(r=0)^n ^nC_rx^r and (1-x)^n= ^nC_0x^0-^nC_1 x^1 +^nC_2-^nC_3x^3+………+(-1)^n ^nC_nx^n=sum_(r=0)^n (-1)^r ^nC_r x^r On the basis of above information answer the following question: If n is a positive integer then 1/((49)^n) - 8/((49)^n)(^(2n)C_1)+8^2/((49)^n)( ^(2n)C_2)- 8^3/((49)^n)(^(2n)c_3)+......+8^(2n)/((49)^n)= (A) -1 (B) 1 (C) (64/49)^n (D) none of these

Prove that C_(0)2^(2)C_(1)+3C_(2)4^(2)C_(3)+...+(-1)^(n)(n+1)^(2)C_(n)=0 where C_(r)=nC_(r)

Show that nC_(r)=(n-r+1)/(r)*(nC_(r-1))

Prove that ^nC_(r)+^(n-1)C_(r)+...+^(r)C_(r)=^(n+1)C_(r+1)