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_(1)+nC_(2)+...+nC_(n)=2^(n)-1

If n is a positive integer, for (1+x)^(n) ,then sum_(r=0)^(n).^nC_(r)=

If n is a positive integer,sum_(r=0)^(n)(^(^^)nC_(r))^(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 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

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

Write the expression ^nC_(r+1)+^(n)C_(r-1)+2xx^(n)C_(r) in the simplest form.

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

Prove that ^nC_(r)+^(n-1)C_(r)+...+^(r)C_(r)=^(n+1)C_(r+1)

^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