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 of the above information answer the following questions 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 of 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 of 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 (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 C_r stands for nC_r, then the sum of first (n+1) terms of the series a C_0-(a+d)C_1+(a+2d)C_2-(a+3d)C_3+......, is

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)

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. 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)

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

Find sum of sum_(r=1)^n r . C (2n,r) (a) n*2^(2n-1) (b) 2^(2n-1) (c) 2^(n-1)+1 (d) None of these

^nC_r+^nC_(r+1)+^nC_(r+2) is equal to (2lerlen) (A) 2^nC_(r+2) (B) 2^(n+1)C_(r+1) (C) 2^(n+2)C_(r+2) (D) none of these

1/2. ""^nC_0 + ""^nC_1 + 2. ""^nC_2 + 2^2. ""^nC_3 + …….+ 2^(n-1) . ""^nC_n =