Prove that `^nC_0^(2n)C_n-^n C_1^(2n-2)C_n+^n C_2^(2n-4)C_n=2^ndot`

Prove that ^nC_(0)^(2n)C_(n)-^(n)C_(1)^(2n-1)C_(n)+^(n)C_(2)xx^(2n-2)C_(n)++(-1)^(n)sim nC_(n)^(n)C_(n)=1

Prove that ^nC_(0)^(n)C_(0)-^(n+1)C_(1)^(n)C_(1)+^(n+2)C_(2)^(n)C_(2)-...=(-1)^(n)

Prove that : ""^(n)C_(0).""^(2n)C_(n)-""^(n)C_(1).""^(2n-2)Cn_(n)+""^(n)C_(2).""^(2n-4)Cn_(n)+......=2^n

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

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

Prove that C_(0)^(2)+C_(1)^(2)+...C_(n)^(2)=(2n!)/(n!n!)

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + …+ C_(n) x^(n) , prove that C_(0)^(2) - C_(1)^(2) + C_(2)^(2) -…+ (-1)^(n) *C_(n)^(2)= 0 or (-1)^(n//2) * (n!)/((n//2)! (n//2)!) , according as n is odd or even Also , evaluate C_(0)^(2) + C_(1)^(2) + C_(2)^(2) - ...+ (-1)^(n) *C_(n)^(2) for n = 10 and n= 11 .

Prove that: ^(2n)C_0-3.^(2n)C_1+3^2.^(2n)C_2-..+(-1)^(2n) ..3^(2n)^(2n)C_(2n)=4^n for all value of N

Prove that (""^(2n)C_(0))^(2)-(""^(2n)C_(1))^(2)+(""^(2n)C_(2))^(2)-…+(""^(2n)C_(2n))^(2)=(-1)^(n)*""^(2n)C_(n) .