Prove that: . n`^C_0+2.^nC_1+…2^n.^nC_n=3^n` for every natural number n.

Prove that: .^(n-1)C_3+^(n-1)C_4gt^nC_3, if ngt7

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

If n is a positive integer such that (1+x)^n=^nC_0+^nC_1+^nC_2x^2+…….+^nC_nx^n, for epsilonR . Also ^nC_r=C_r On the basis o the above information answer the following questions for any aepsilon R the value of the expression a-(a-1)C_1+(a-2)C-2-(a-3)C_3+.+(1)^n(a-n)C_n= (A) 0 (B) a^n.(-1)^n.^(2n)C_n (C) [2a-n(n+1)[.^(2n)C_n (D) none of these

If n is a positive integer such that (1+x)^n=^nC_0+^nC_1+^nC_2x^2+…….+^nC_nx^n, for epsilonR . Also ^nC_r=C_r On the basis o the above information answer the following questions The value of the series sum_(r=1)^n r^2.C_r= (A) 1 (B) (-1)^(n/2).n!/(n/2!)^2 (C) (n-1).^(2n)C_n+2(2n) (D) n(n+1)2^(n-2)

If n is a positive integer such that (1+x)^n=^nC_0+^nC_1+^nC_2x^2+…….+^nC_nx^n, for epsilonR . Also ^nC_r=C_r On the basis o the above information answer the following questions the value of ^mC_r.^nC_0+^mC_(r-1).^nC_1+^mC_(r-2).^nC_2+….+^mC_1.^nC_(r-1)+^nC_0^nC_r where m,n, r are positive interges and rltm,rltn= (A) ^(mn)C_r (B) ^(m+n)C_r (C) 0 (D) 1

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

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-...+(-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)

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