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 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

Find the sum sum_(r=1)^n r^n (^nC_r)/(^nC_(r-1)) .

Prove that sum_(r = 0)^n r^2 . C_r = n (n +1).2^(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)) )

Deduce that: sum_(r=0)^n .^nC_r (-1)^r 1/((r+1)(r+2)) = 1/(n+2)

sum_(r=1)^(n) (-1)^(r-1) ""^nC_r(a - r) =

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 sum_(r = 1)^n r^3 ((n_C_r)/(C_(r - 1)))^2 = (n (n + 1)^2 (n+2))/(12)