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: Coefficient of `x^50 in (1+x)^1000+x(1+x)^999+........+x^999(1+x)+x^1000 ` is (A) `^1000C_50` (B) `^1002C_50` (C) `^1001C_50` (D) `^1001C_49`

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

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:Um of coeficients of x^50 and x^51 in (1+x)^199+(1+x)^198x+(1+x)6197x^2+..+(1+x)x^198+x^199 is euqla to the coefficient of x^r in (1+x)^200+(1+x)^199x+(1+x0^198x^2+.........+(1+x)x^199+x^200 then r may be equal to (A) 51 (B) 52 (C) 53 (D) none of these

Find the coefficients of x^(50) in the expression (1+x)^(1000)+2x(1+x)^(999)+3x^(2)(1+x)^(998)+...+1001x^(1000)

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 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)

Find the coefficient of x in polynomial (x+2n+1C_(0))(x+^(2n+1)C_(1))......(x+^(2n+1)C_(n))

The coefficients of x^n in the expansion of (1+x)6(2n) and (1+x0^(2n-1) are in the rtio of (A) 1:2 (B) 1:3 (C) 3:1 (D) 2:1

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