" XAMPLE 18.Prove that "C_(0)^(2)+C_(1)^(2)+C_(2)^(2)+...+C_(n)^(2)=(2n!)/((n!)^(2))

prove that :C_(0)^(2)+3C_(1)^(@)+5C_(2)^(2)+...+(2n+1)C_(n)^(2)=((n+1)2n!)/((n!)^(2))

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

Prove that ^nC_(0)^(2n)C_(n)-^(n)C_(1)^(2n-2)C_(n)+^(n)C_(2)^(2n-4)C_(n)-...=2^(n)

Prove that 3.C_(0)^(2)+7.C_(1)^(2)+11.C_(2)^(2)+…..+(4n+3).C_(n)^(2)=(2n+3).""^(2n)C_(n).

Prove that C_(1)^(2)-2*C_(2)^(2)+3*C_(3)^(2)-…-2n*C_(2n)^(2)=(-1)^(n)n*C_(n)

Prove that C_(1)^(2)-2*C_(2)^(2)+3*C_(3)^(2)-…-2n*C_(2n)^(2)=(-1)^(n)n*C_(n)

Prove that : For n = 0, 1, 2, 3, ………., n, prove that C_(0).C_(r)+C_(1).C_(r+1)+C_(2).C_(r+2)+….+C_(n-r).C_(n) =""^(2n)C_((n+r)) and hence deduce that Prove that : C_(0)^(2)+C_(1)^(2)+C_(2)^(2)+….+C_(n)^(2)=""^(2n)C_(n)

Prove that C_(0)^(2)-C_(1)^(2)+C_(2)^(2)-C_(3)^(2)+….+(-1)^(n).C_(n)^(2)={{:((-1)^(n//2)""^(n)C_(n//2)",","if n is even"),(" "0" ,","if n is odd"):}

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 .

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 .