Prove that: `2xx^7C_4=^8C_4`

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

Verify that 2 ""^(7)C_(4)=""^(8)C_(4) .

Prove that (1)8C_(0)+8C_(1)+8C_(2)+...+8C_(8)=256...(2)9C_(0)+9C_(2)+9C_(4)+......+9C_(8)=256

Verify that if 2xx ""^(7)C_(4)= ""^(17)C_(8) .

Prove that : ""^(2)C_(1)+ ""^(3)C_(1)+""^(4)C_(1)=""^(3)C_(2)+""^(4)C_(2) .

If a,b,c are in A.P.,prove that: (a-c)^(2)=4(a-b)(b-c)a^(2)+c^(2)+4ac=2(ab+bc+ca)a^(3)+c^(3)+6abc=8b^(3)

If ={2,3},B={1,2,3} and C={2,3,4} Prove that A xx A=(B xx B)nn(C xx C)

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 .

Theorem 8( For any set A;B and C prove that A xx(B'uu C')'=(A xx B)nn(A xx C)