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` (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 such that (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)) omega + (C_(1) + C_(4) + C_(7) + C_(10) + C_(16)) omega^(2) = 0 , where omega is a cube root of unity .

C_0C_2 + C_1C_3 +C_2C_4+……..+C_(n-2) C_n =

C_0 + C_1 + 2.C_2(3) + 3.C_3(3^2)+ 4.C_4(3^3) + ……+n.C_n 3^(n-1) =

If (1+x)^n = C_0 + C_1x + C_2x^2 + ……..+C_n.x^n then find C_0 - C_2 + C_4 - C_6 + …….

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) + …

(C_0)/(1) + (C_2)/(3) + (C_4)/(5) + ……+(C_16)/(17) =

(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

Prove that C_0.C_3 + C_1.C_4 + C_2.C_5 + …..+C_(n-3).C_n = ""^(2n)C_(n +3)

If (1 + x + x^2)^n = C_0 + C_1x + C_2x^2 + C_3x^3 + ……..C_nx^n then find C_0 + C_3 + C_6 + …….

(C_(1))/(C_(0))+(2.C_(2))/(C_(1))+(3.C_(3))/(C_(2))+ . . .+(20.C_(20))/(C_(19))=