Prove that `C(24,5)+C(24,6)+C(25,7)+C(26,8)+C(27,9)+C(28,10)=C(29,19)`

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 .

Prove that .^(7)C_(2)+^(7)C_(3)=^(8)C_(3)

""^(25)C_(22)- ""^(24)C_(21)= …………

Prove that : ""^(25)C_(10)+""^(24)C_(10)+……..+""^(10)C_(10)=""^(26)C_(11)

Prove that : ""^(25)C_(10)+""^(24)C_(10)+……..+""^(10)C_(10)=""^(26)C_(11)

Prove that ""^8C_5+""^8C_4=""^9C_5

Prove that , .^(15)C_(8)+^(15)C_(9)-.^(15)C_(6)-^(15)C_(7)=0

Prove that ""^(24)C_(4)+overset(4)underset(r=0)sum^((28-r))C_(3)=^(29)C_(4)

5/8+3/4-7/(12) is equal to: (a) (15)/(24) (b) (17)/(24) (c) (19)/(24) (d) (21)/(24)