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

If n is a positive integer,then nC_(1)+nC_(2)+...+nC_(n)=2^(n)-1

If n and r are integers such that 1

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

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 ^nC_r+^nC_(r+1)=^(n+1)^C_(r+1) Also coefficient of x^r in the expansion of (1+x)^n=^nC_r. In an identity in x, coefficient of similar powers of x on the two sides re equal. On the basis of above information answer the following question: If n is a positive integer then ^nC_n+^(n+1)C_n+^(n+2)C_n+.....+^(n+k)C_n= (A) ^(n+k+1)C_(n+2) (B) ^(n+k+1)C_(n+1) (C) ^(n+k+1)C_k (D) ^(n+k+1)C_(n-2)

For all positive integers n, show that ^2nC_(n)+^(2n)C_(n-1)=(1)/(2)(^(2n+2)C_(n+1))

Sum of the series a^n+a^(n-1)b+^(n-2)b^2+………..+ab^n can be obtained by taking outt a^n or b^n comon and using the forumula of sum of (n+1) terms of G.P. N the basis of above information answer the following question:Coefficientoif xp, (0leplen) in 3^(n-1)+3^(n-2)(x+3)+3^(n-3)(x+3)^2+............+(x+30)^(n-1) is (A) ^nC_p3^n-p) (B) ^(n+1)C_p3^(n-p+1) (C) ^nC_p3^(n-p-1) (D) none of these

The middle term in the expansion of ((2x)/(3)-(3)/(2x^(2)))^(2n) is ^(2n)C_(n) b.(-1)^(n)2nC_(n)x^(-n) c.^(2n)C_(n)x^(-n)d .none of these