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)

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 and C_r = ""^nC_r then find the value of sum_(x = 1)^n r^2 ((C_r)/(C_(r - 1)) )

If n is an odd natural number, then sum_(r=0)^n (-1)^r/(nC_r) is equal to

The coefficient of x^(4) in the expansion of (1+x+x^(2)+x^(3))^(11) is where ({:(n),(r):}) = nC_(r)

Prove that coefficient of x^(r ) in (1-x)^(-n) " is " ""^(n+r-1)C_(r )

Given positive integers r>1,n> 2, n being even and the coefficient of (3r)th term and (r+ 2)th term in the expansion of (1 +x)^(2n) are equal; find 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 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 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 ^nC_r=84 ,^n C_(r-1)=36 ,a n d^n C_(r+1)=126 , then find the value of ndot