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Prove that C0^2+C1^2+...Cn^2=(2n!)/(n! n...

Prove that `C_0^2+C_1^2+...C_n^2=(2n!)/(n! n!)`

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If C_0, C_1,C_2………..C_n be the coefficients of expansion (1+x)^n prove that C0C2+C1C3+C2C4+C_n2n= 2n! /(n-2)! (n+2)!

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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 .

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Prove that "^(2n)C_0 + ^(2n)C_2 + .... + ^(2n)C_(2n) = 2^(2n-1)

Prove that C _(0) +2 . C _(1) + 2 ^(2) . C _(2) + .......+ 2 ^(n) . C _(n) = 3 ^(n).