Prove that `sum_(r=0)^n r(n-r)C r2=n^2(^(2n-2)C_n)dot`

Prove that sum_(r=0)^(2n)(.^(2n)C_(r))^(2)=n^(4n)C_(2n)

Prove that sum_(r=0)^(n)C_(r)sin rx cos(n-r)x=2^(n-1)sin(nx)

Prove that sum_(n)^(r=0) ""^(n)C_(r)*3^(r)=4^(n).

If (1+x)^n=sum_(r=0)^n C_rx^r then prove that sum_(r=0)^n (C_r)/((r+1)2^(r+1))=(3^(n+1)-2^(n+1))/((n+1)2^(n+1))

Prove by mathematical induction that sum_(r=0)^(n)r^(n)C_(r)=n.2^(n-1), forall n in N .

Statement -1: sum_(r=0)^(n) r(""^(n)C_(r))^(2) = n (""^(2n -1)C_(n-1)) Statement-2: sum_(r=0)^(n) (""^(n)C_(r))^(2)= ""^(2n)C_(n)

The value of sum_(r=0)^(n) r(n -r) (""^(n)C_(r))^(2) is equal to