Prove that `^100 C_2^(100)C_2+^(100)C_2^(100)C_4+^(100)C_4^(100)C_6++^(100)C_(98)^(100)C_(100)=1/2[^(200)C_(98)-^(100)C_(49)]dot`

Prove that .^100 C_0^(100)C_2+^(100)C_2^(100)C_4+^(100)C_4^(100)C_6++^(100)C_(98)^(100)C_(100)=1/2[.^(200)C_(98)-^(100)C_(49)]dot

Prove that ^100 C_0^(100)C_2+^(100)C_2^(100)C_4+^(100)C_4^(100)C_6++^(100)C_(98)^(100)C_(100)=1/2[^(200)C_(98)-^(100)C_(49)]dot

Prove that ^100C_(2)^(100)C_(2)+^(100)C_(2)^(100)C_(4)+^(100)C_(4)^(100)C_(6)+...+^(100)C_(98)^(100)C_(100)=(1)/(2)[^(200)C_(98)-^(100)C_(49)]

Prove that 100C_(0),100C_(2)+100C_(2),100C_(4)+100C_(4)+100C_(6)+....+100C_(98),100C_(100)=(1)/(2)[200C_(98)-100C_(49)

Prove the following identieties using the theory of permutation where C_(0),C_(1),C_(2),……C_(n) are the combinatorial coefficents in the expansion of (1+x)^n,n in N: ""^(100)C_(10)+5.""^(100)C_(11)+10 .""^(100)C_(12)+ 10.""^(100)C_(13)+ 10.""^(100)C_(14)+ 10.""^(100)C_(15)=""^(105)C_(90)

Evaluate: (i) ^(10)C_3 (ii) ^(100)C_(98) (iii) ^(60)C_(60)

Sum of the series (100C_(1))^(2)+2.(100C_(2))^(2)+3.(100C_(3))^(2)+....+100(100C_(100))^(2) equals

The coefficient of x^(53) in the expansion sum_(m=0)^(100)^(100)C_m(x-3)^(100-m)2^m is ^100 C_(47) b. ^100 C_(53) c. -^(100)C_(53) d. none of these