If for `z` as real or complex, `(1+z^2+z^4)^8=C_0+C1z2+C2z4++C_(16)z^(32)t h e n` `C_0-C_1+C_2-C_3++C_(16)=1` `C_0+C_3+C_6+C_9+C_(12)+C_(15)=3^7` `C_2+C_5+C_6+C_(11)+C_(14)=3^6` `C_1+C_4+C_7+C_(10)+C_(13)+C_(16)=3^7`

If for z as real or complex,(1+z^(2)+z^(4))^(8)=C_(0)+C1z2+C2z4++C_(16)z^(32)then(a)C_(0)-C_(1)+C_(2)-C_(3)+C_(16)=1(b)C_(0)+C_(3)+C_(6)+C_(9)+C_(12)+C_(15)=3^(7)(c)C_(2)+C_(5)+C_(6)+C_(11)+C_(14)=3^(6)(d)C_(1)+C_(4)+C_(7)+C_(10)+C_(13)+C_(16)=3^(7)

If for z as real or complex . (1+z^(2) + z^(4))^(8) = C_(0) C_(1) z^(2) C_(2) z^(4) + …+ C_(16) z^(32) , prove that C_(0) + C_(3) + C_(6) + C_(9) + C_(12) + C_(15) + (C_(2) + C_(5) + C_(8) + C_(11) + C_(14)) + (C_(1) + C_(4) + C_(7) + C_(10) + C_(16)) omega^(2) = 0 , where omega is a cube root of unity .

C_(0)C_(1)+C_(1)C_(2)+C_(2)C_(3)+...+C_(n-1)C_(n)

C_(0)C_(2)+C_(1)C_(3)+C_(2)C_(4)+c_(3)C_(5)+...+C_(n-2)C_(n)

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + C_(3)x^(3) + …+ C_(n) x^(n) , then C_(0) - (C_(0) - C_(1)) + (C_(0) + C_(1) + C_(2))- (C_(0) + C_(1) + C_(2)+ C_(3)) + ...+ (-1)^(n-1) (C_0) + C_(1) + C_(2) + ...+ C_(n-1)) , when n is even integer is

If (1 + x)^(n) = C_(0) + C_(1)x + C_(2)x^(2) + C_(3) x^(3) + C_(4) x^(4) + ..., find the values of (i) C_(0) - C_(2) + C_(4) - C_(0) + … (ii) C_(1) - C_(3) + C_(5) - C_(7) + … (iii) C_(0) + C_(3) + C_(6) + …

(1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + C_(3) x^(3) + … + C_(n) x^(n) , prove that C_(0) - 2C_(1) + 3C_(2) - 4C_(3) + … + (-1)^(n) (n+1) C_(n) = 0