If `C_r` stands for `nC_r`, then the sum of the series `(2(n/2)!(n/2)!)/(n !)[C_0^2-2C_1^2+3C_2^2-........+(-1)^n(n+1)C_n^2]` ,where n is an even positive integer, is

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

The sum of the series sum_(r=0) ^(n) ""^(2n)C_(r), is

sum_(r=1)^k(−3)^(r−1) .^(3n)C_(2r−1)=0, where k=(3n)/2 and n is an even integer

C_1/C_0+2C_2/C_1+3C_3/C_2+............+nC_n/C_(n-1)=(n(n+1))/2

Prove that (^(2n)C_0)^2-(^(2n)C_1)^2+(^(2n)C_2)^2-+(^(2n)C_(2n))^2-(-1)^n^(2n)C_ndot

Show that C_1^2-2C_2^2+3.C_3^2- ……….-2n.C_(2n)^2=(-1)^(n-1).n.C_n where C_r stands for ''^(2n)C_r''

If "^(n)C_(0)-^(n)C_(1)+^(n)C_(2)-^(n)C_(3)+...+(-1)^(r )*^(n)C_(r )=28 , then n is equal to ……

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^(2) (""^(n)C_(r))/(""^(n)C_(r-1)) .

Prove that "^n C_0^(2n)C_n-^n C_1^(2n-1)C_n+^n C_2xx^(2n-2)C_n++(-1)^n^n C_n^n C_n=1.